I think that (1) is a tautology whereas no evidence has been offered in support of (2) — Michael
It is not built from anything. (0,1) is one object - a line — keystone
It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone
Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths: — keystone
you and fryfish are having a really tough time — keystone
And don't mix philosophy of mathematics with the real deal. — jgill
I don't understand why you would say this — keystone
Do you think I'm using the term topological incorrectly? — keystone
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
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.
Ditto.Ok. My eyes glaze a little more every time you mention the S-B tree — fishfry
the metric space is topological — keystone
WikipediaFormally, 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.
I have a strong affinity for the Stern-Brocot Algorithm — keystone
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
?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
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
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
Since you're joining in on the conversation, can you tell me if anything I'm saying makes sense to you? — keystone
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
Step two entails describing these intervals within the framework of a topological metric space. — keystone
More like a inspiration to get peoples own ideas going — Elnathan
The math doesn't identify any particular stopping point, but it does imply there has to be one. — Relativist
However: the kinematic process never actually reaches time 1 — Relativist
By definition, a limit is not reached, — Relativist
However, the clock does reach 1. At time 1, the stairway descent must have ended — Relativist
I would say they were very intelligent. — isomorph
?a thinking capacity equal to ours, maybe greater than ours
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
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
I decided that I will contact the philosophy professor at the local college and ask him if he can be my mentor. — Fermin
How might all these dynamics interact? — Benj96
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
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
**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
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
Can a brutha get an AMEN?! :sweat: :up: — 180 Proof
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