It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone

Unfortunately, your approach is a muddled mixture of traditional ideas and speculative continua. I think you need to go back to the beginnings of your efforts and truly start with continua and develop an original approach to math in which points arise from these continua, avoiding the real line entirely at first. MU has written a similar notion about continua and points. Perhaps you can put some meat on the table.

Define a continuum as an abstract entity and not in terms of the real line. As a matter of fact, use another word for your creation. State the properties of the continuum, again not referencing the real line or numbers. This is a tall order. Metric spaces and topological functions are perhaps inappropriate in this regard. I don't know. You will be going into unexplored territory.

Please stop talking about the S-B algorithm. Both my colleague and I would rather not contemplate this thing. Leave the realm of real numbers at first.

Or, do your thing and persist until the thread dries up and vanishes. Good luck.

Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths: — keystone

It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions.
How does this affect your metric?

you and fryfish are having a really tough time — keystone

:cool:

And don't mix philosophy of mathematics with the real deal. — jgill


I don't understand why you would say this — keystone

Very very few contemporary mathematicians give a fig leaf about Platonic vs non-Platonic arguments or similar discussions about whether math is embedded in nature or in the mind.

Do you think I'm using the term topological incorrectly? — keystone

Well, if you were to avoid both metric spaces and variations of the word "topology" it might mitigate what seems to be a questionable attempt to employ legitimate mathematical notions within a somewhat murky mix of ideas. However, I applaud your enthusiasm. I used to teach point set topology and metric spaces, so I am biased toward their traditional interpretations. In any event at some point you must present a clear and detailed description of your ideas that mathematicians might have reservations about but can follow the logic.

What would be a homeomorphism of [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] ?

The late George Simmons of Colorado College wrote a marvelous book many years ago, perhaps the finest introduction to modern analysis and topology ever written: Introduction to Topology and Modern Analysis. Slowly work your way through this book and you will see why we ask so many questions. And don't mix philosophy of mathematics with the real deal. Just an I idea.

Intuitionism is closely related to constructivism, the idea that mathematical objects only exist if there's an algorithm or procedure to construct them. Intuitionism is like constructivism with an extra bit of mysticism that I can never quite grasp. — fishfry

On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:

Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

Ok. My eyes glaze a little more every time you mention the S-B tree — fishfry

Ditto.

the metric space is topological — keystone

So, a continuous deformation takes path A to path B, but inside the ms of path A? Or a new ms of path B? You might illustrate this. I'm curious about these continuous deformations in the contexts of your ideas. A topology, on the other hand . . .

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
Both the empty set and X are elements of τ. Any union of elements of τ is an element of τ. Any intersection of finitely many elements of τ is an element of τ.
If τ is a topology on X, then the pair (X, τ) is called a topological space.

Wikipedia

The saddest words of tongue or pen: it might have been.

I have a strong affinity for the Stern-Brocot Algorithm — keystone

This is perhaps the second time this oddity from number theory has cropped up on this forum. I knew a tiny bit about it since it can involve elementary continued fraction theory. How did this become so important to you?

Is this a legitimate "path" ?. A linear arrangement of points and intervals.

Ordered pair 1: (3,3)
Ordered pair 2: (3,3.7)
Ordered pair 3: (3.7,3.7)
Ordered pair 4: (3.7,4.2)
Ordered pair 5: (4.2,4.2)
Ordered pair 6: (4.2,5.1)
Ordered pair 7: (5.1,5.1)

So, a "metric space" for this path consists of "points" (a,b) within this structure. For example, d((3,3.7),(4.2,5.1))=|(3+3.7)/2 - (4.2+5.1)/2| = 1.3. So you do not compare "points" from one path to another. Altering the path, even slightly, places it in another metric space. But a ms could be a subspace of a bigger ms. Just talking to myself, here.

You spoke earlier of an "elastic band". where does that come into the picture? Especially with regard to metric spaces? Can a path be circular?

d([2,3],[1,4])=0 ? [2,3] not equal to [1,4] — jgill

Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set — keystone

You speak of a metric space. Precisely what are the "points" in such a space? Then explain the metric you have created giving "distances" between these points.

From the Bleachers: Does

$[\pi ,\pi ]\cup (\pi ,\pi +1)\cup [\pi +1,\pi +1]\cup (\pi +1,5)\cup [5,5]$

exist in your system? Or are you assuming rational numbers only?

Is

$[5,5]\cup (5,5+\varepsilon )\cup [5+\varepsilon ,5+\varepsilon ]\cup (5+\varepsilon ,5+2\varepsilon )\cup [5+2\varepsilon ,5+2\varepsilon ]\approx 5$

for small epsilon?

Step 5: Arrive at point 0. — keystone

?

Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set — keystone

I don't know what you are talking about. You and @fishfry can sort all of this out if he is willing. Good luck.

"In a metric, the distance between two distinct points as always positive."


d([2,3],[1,4])=0 ? [2,3] not equal to [1,4].

According to some interpretations of irrational Infinity though, an infinite-sided die is not impossible, only supernatural, in the sense that you can imagine it, as an ideal concept --- e.g. a perfect multidimensional sphere --- but never reach-out and grasp it, in the real world. In what sense does that set of one "imaginary die" exist? :joke: — Gnomon

What is "irrational infinity"? Infinite sided die seems like a sphere in 3D.

Mathematics has a similar structure to certain conceptions of magic. It requires years of studying something entirely incorporeal, it seems to exist independent of the physical realm, it’s very powerful and has the ability to predict and influence the world around us, and it’s practitioners are BIZARRE — Gnomon

Does magic influence the world around us? Wow, what bizarre powers I wield! :cool:

Since you're joining in on the conversation, can you tell me if anything I'm saying makes sense to you? — keystone

Not much, I fear. But I stand aside and try not to impede the ongoing discussion between you and @fishfry, who is more familiar with modern math topics that I am. I am curious about the role of elasticity you describe regarding one of Zeno's paradoxes. I assume this is somehow basic to where you are headed.

As I mentioned before, the metric between ordered pairs (x1,y1) and (x2,y2) is defined as follows: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 | — keystone

Is (x1,y1) a point in the Euclidean plane? That's a standard designation. Then (x1+y1)/2 requires some kind of a projection of the y1 down upon the x-axis in order for this expression to represent a midpoint of a line segment. Or, is (x1,y1) an interval on the real line? Perhaps use (a1,b1) instead to keep the level of confusion minimized. Maybe my colleague sees something here I don't. Please carry on.

Not so sure philosopher and critical thinker are one and the same.

Step two entails describing these intervals within the framework of a topological metric space. — keystone

You have used this expression frequently. Do you know what you are talking about? Just curious.

More like a inspiration to get peoples own ideas going — Elnathan

Works sometimes. Not always. A brief, concise OP helps.

The math doesn't identify any particular stopping point, but it does imply there has to be one. — Relativist

An exercise in free will. At each n the cube changes from white to black or vice-versa for time = 1-1/2^n. The clock runs out so you are free to say the process has ended and the final color is black. Your friend can say, no it is white. But you will prevail.

However: the kinematic process never actually reaches time 1 — Relativist

That's no surprise. It is an imaginary contrivance impossible to physically fabricate. Just a thought.

From the bleachers: Hasn't MU spoken of the existence of continua prior to points? What of Bergson's notion of time being fundamentally duration? Perhaps not elastic. Is this where the discussion is going? Peter Lynds has written several papers on the non-existence of points in time, as well.

Back to the popcorn.

Non-understanding does not equal understanding. — Echogem222

No argument with that. :roll:

For those, like me, who didn't know: Fallacy of the undistributed middle

Pass the popcorn, please. I am sitting in the bleachers watching with interest. :chin:

By definition, a limit is not reached, — Relativist

$f(x)=3x+2$, $f(1)=5$

$\underset{x\to 1}{\mathop{\lim }}\,f(x)=f(1)$

However, the clock does reach 1. At time 1, the stairway descent must have ended — Relativist

Certainly the relationship between time (independent of human control) and physical steps taken over a period of time has ended.

I would say they were very intelligent. — isomorph

But

a thinking capacity equal to ours, maybe greater than ours

?

Welcome to the Forum. :cool:

Our prehistorical ancestors had thinking capacity equal to ours, maybe greater than ours, and this can be seen in prehistoric cave art created by intellectual masters. — isomorph

How do the drawings on cave walls imply intellectual capacities rather than simply artistic abilities?

So there's just no paradox. There is only taking perfectly well-understood mathematical facts and dressing them up with physics-contradicting staircases and lightbulbs so as to confuse people. — fishfry

:up: Amen :roll:

I decided that I will contact the philosophy professor at the local college and ask him if he can be my mentor. — Fermin

I hope that works for you. I was a math prof for many years, and at large universities senior faculty might simply give an unsolicited paper to a grad student as an assignment to critique. It may be the same in philosophy.

How might all these dynamics interact? — Benj96

I wonder how dynamic systems would function. At each step there would be many options.

Wonder what that 8,500 crossings/day means? Not enough energy to check it out.

I say yes. I predict that in my area, math, there will be computer devised concepts and results beyond human understanding. If simply the length of a particular discovery - only so much that can be kept in the mind and be instantly retrievable for comparison.

Maybe not.

While they may not regard him as The Messiah the do believe his is a messiah and like all messiahs persecuted by the enemies of God. — Fooloso4

Or perhaps the only apparent option to remove the Biden cabal. Don't forget, Joe told immigrants to "Surge the Border", and gave Afghanistan back to the Taliban - where recently the supreme Poohbah stated he is bringing back stoning certain women to death.

The only problem with this type of exercise is that the dog does a lot of sniffing and we don't move very fast — Agree-to-Disagree

Be tolerant and thankful for a furry companion. :cool:

**Description**: Apply ethical theories to real-world scenarios. Discuss topics like environmental ethics, personal and social justice, and personal and social responsibility.
- **Age Appropriateness**: Middle schoolers are forming their identities and need guidance on ethical decision-making. — Lif3r

Add The Philosophy of Taylor Swift to the middle school part.

Finally got it down to a decent routine. Here’s my current schedule (in case you want to doze off — Mikie

Good for you. Keep it up. When you are very old something may knock you down, then as you try to recover, doctors will find something else. A downward spiral difficult to pull out of.

But I have not found a single one who will even respond to any question as to anyone who should not have a gun. — tim wood

I tend to vote more conservative than liberal these days and I object to illegal immigrants having guns.

Can a brutha get an AMEN?! :sweat: :up: — 180 Proof

No amen yet. Still hoping for a different Dem ticket. "Surge the Border Biden" deserves defeat.

If most citizens have the right to gun ownership, what of illegal migrants?

I'm proceeding with the assumption A.I. will be overtaking the task of heavy lifting re: thought. I'm rooting for S.A.I. in our lifetimes to run up cognitive yardage pushing past what human can imagine. Wittgenstein has directed our attention towards "the silence," conjecture unimaginable. Its nigh time for The Oracle: SAI to start sending us revelations from Wittgenstein's principled imagination silenced. We won't understand but a fraction of the import of the messages, but we'll get pushed to our utter limitations before being back-numbered into the subordinate section of the evolution hierarchy. — ucarr

:up:

It won't be long before mathematical concepts and results stretch beyond our abilities to comprehend.

If you take pi or the golden ratio or eulers number for example, eventually it will detail your entire genetic sequence from start to finish — Benj96

Reference?

Do you condone dishonesty and pretension? — Corvus

When it amuses. Keep going. :cool: