Set theory is needed for the rest of math and so is logic — Srap Tasmaner
But here I would question whether the notion of cause adds anything that is not already given in the mechanistic description. — SophistiCat
You seem to want to dilute the concept so as to include just about any kind of mechanistic analysis, which is tantamount to eliminating causation — SophistiCat
But how is that "checking the validity of one argument using another"? — TonesInDeepFreeze
I don't know what you mean. Example? — TonesInDeepFreeze
https://en.wikipedia.org/wiki/Proof_by_contradiction — Hanover
Regarding knowledge of the curriculum - the kind of student that satisfies all benchmarks in a subject has already been discouraged from pursuing their insights and skills due to herding them through the curriculum at a required pace. It is not uncommon to see an allegedly failing student have a profound insight, which you nevertheless cannot spend time developing with them — fdrake
But since my big post from 6 days ago was opaque to you I'm quite discouraged from continuing on. — keystone
Not sure. But a lot of potential PhDs seem to wash out just from the long dreariness and uncertain job prospects after completion, — BC
When you mention "the Lounge," are you referring to an actual place, or do you mean taking a pause in the conversation until others join in? — keystone
Overall, I think you have started down a path that is far too complicated for the desired result. — jgill
Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. — keystone
I think there's also a difference between "memory" or memorizing something and the knowledge (and the understanding) of something. — ssu
I would like to ask the reader about how does the reader suppose that knowledge can influence one's identity? — Shawn
A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two potential points that are directly connected to it. — keystone
It's a philosophical question which most philosophers are not equipped to even begin to answer — flannel jesus
But what's that got to do with the topic of the thread? — Clearbury
DKE is accurately characterized as 'the stupider a person is, the less likely they are to realize how stupid they are' — Clearbury
the main point is that the DKE is accurately characterized as 'the stupider a person is, the less likely they are to realize how stupid they are' — Clearbury
i wouldn't have thought an expert would write a wikipedia page - they're too busy being experts — Clearbury
However, in 2D and higher dimensions, a curve is determined not only by its endpoints but also by an equation. Perhaps incorporating that equation into the vertex might make the concept more digestible — keystone
And isn't Wikipedia written by those who fancy themselves experts in matters they have no expertise on? — Clearbury
I see it averages about 47 pageviews per day on Wiki, and classed as low priority. — jgill
Sometimes the significance of a discovery isn't recognized until many years later. — keystone
Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality — keystone
(Wiki)From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects.
A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, pseudo point, or actual curve — keystone
I’m a bit surprised that once I introduced a more mathematical approach—like discussing the Stern-Brocot tree and providing proper definitions—you felt the discussion was becoming less interesting to mathematicians. I had expected the opposite. — keystone
All deterministic processes are time-symmetric — SophistiCat
There's an elegance to QM and I believe the same can be said about the top down view of mathematics — keystone
If you had two functions on Q then a suitable metric would be the supremum. — jgill
Suitable for what? — keystone
Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so. — jgill
I've tried in the past, but nowhere else has been as beneficial as here — keystone
For example, when I write "n∈ N", I don’t mean that n is an element of the actual infinite set of natural numbers. Rather, I mean that, it is a natural number according to the SB tree (details omitted). — keystone
The function x(n) — keystone
An actual curve is an indivisible, one-dimensional object with length but no width or depth. It extends continuously between two actual points but excludes the endpoints. — keystone
Too bad that the basics of category theory aren't taught in school. But then again, the educational system doesn't care much about the philosophy of mathematics or the foundations of mathematics. — ssu
I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. — keystone
As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill
Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like R? — keystone
But that program (even in an infinite world*) cannot actually output a set with a cardinality of ℵ0. Potential is important and I feel like it's been forgotten in our Platonist world. — keystone
I think the whole idea grossly overestimates people's interest in having an opinion on every political subject all the time — Benkei
By moving the focus from the destination to the journey the need for actual infinity vanishes. — keystone
In my recent posts, I have been establishing that real numbers instead describe potential k-curves, which can be thought of as yet to be constructed k-curves which when constructed have the potential to be arbitrarily small (but always retain a non-zero length). — keystone
Of course it is true you're not obliged to find that interesting. — Wayfarer
The challenge for scientific realism is the concept of superposition — Wayfarer
Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.
He ends up advocating (maybe just "showing the benefits of" is a better term) of an approach grounded in category theory. — Count Timothy von Icarus
Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system.