I'll go one step further. A deeper question is whether the spectrum of reality is continuous. As Einstein inferred, the moon exists - and our imaginations exist. What is in between?

If you say that logic is not merely symbol manipulation, then what do you say it is? — Leontiskos

It's mathematics without the math. :roll:

the paradox of your notes is that whereas the figure collapses uniformly, the surface areas of the 3D figures are limitless. — javi2541997

:up:

I think this recent move by the US to allow Ukraine to use US arms to strike targets deep inside Russia blatantly shows their escalatory intentions. — Tzeentch

11,000 to 100,000 North Korean soldiers muddies the political waters. These special ops troops infiltrate and assassinate.

Nevertheless, this madness must end at the negotiation table.

Depends on what "get with" means. After my first divorce one of my mature students showed up on the front row with an apple on her desk. Later that day she popped into my office. I asked "Where is my apple?". She said, "I ate it". We were happily married for 33 years.

For example, I am not an expert on mathematics, but some paradoxes are interesting, and I want to expand my knowledge of that — javi2541997

Here is a brief note with images on the diagonal paradox, which some on this site argue is not really a paradox. Interestingly, a physicist on this site mentioned it might have relevance in describing or explaining quantum phenomena.

. . . or that contemporary philosophers in general are not interested in mankind’s search for meaning? — Joshs

Perhaps that task has been relocated in psychology and psychiatry. Or where its been for eons, religion.

What I mean about the difficulty of contemporary analytic philosophy, is that it's often extremely dense, written by and for those who can draw on a great deal of specialised scholarship — Wayfarer

Certainly in logic and foundations of mathematics this is true. "Clarity of thought", as Wikipedia states, may arise from the use of well defined symbols and operations thereupon rather than poorly defined words.

If I were a professor, I would evaluate more the grammar than the content itself. Maybe a student is great in math, but if his grammar is terrible, I think he should not be able to promote. Simple. — javi2541997

Where I taught we went through a gentler version of this, grading partly on grammar. But trying to teach both subjects simultaneously was counter productive. And I am not sure the ability to write lengthy pieces with delightful descriptive flourishes is an admirable trait where succinct, to the point passages would suffice - in fact, be more readable. For example, many long convoluted posts made on this site.

because on many philosophical views the goal of a philosophy teacher is not going to be publication, but teaching (really more mentoring), which of course certainly happens, but in academia there is the whole "publish or perish" thing that can often backload this. — Count Timothy von Icarus

We had a small anthropology/philosophy department where I taught. Actually, if I recall, one or two philosophers.

In the larger universities, especially in the sciences, the reason the institutions put pressure on faculty is to get grants that, for instance, release the member from teaching to do research and procure more grants, with the school sharing the incoming money. I see there are several large grants available to philosophical studies, including theology. But probably little coming from the department of defense where the big bucks reside. Correct me if I am wrong.

So Publish or Parish while enhancing an institution's reputation has a stronger motive.

Frankly, I am surprised there are as many grants for philosophy as there are.

I don't see why it needs such a presupposition. Humans have found that nature is intelligible. — Janus

Interesting. Does nature include quantum mechanics and consciousness? — Tom Storm

Give it time and it might explain these phenomena.

Elementary calculus does not require "actual" infinities. — jgill

Calculus uses infinite sets on day one. Even before a student gets to calculus, with analytical geometry we're using infinite sets. The real line and the real plane are infinite sets.

But it seems you mean that calculus doesn't usually mention transfinite ordinals (though the set of natural numbers is a transfinite ordinal), which is true. — TonesInDeepFreeze

That's what I meant. To think of infinity itself as some sort of number. Trying to keep the discussion at an amateur level.

I have used R, but not a transfinite number. Unless I occasionally use the "point at infinity" in complex analysis. — jgill

Points at infinity are not required to be transfinite numbers. — TonesInDeepFreeze

I'm thankful for that. Why resurrect this dead thread?

If there is an air of insubstantiality about this thread, it is because it is concerned with the philosophy of philosophy. This makes it particularly liable to disappear up its own fundament.

Accordingly, I propose, firstly, that philosophy is always parasitic; one might try a 'philosophy of nothing', but it wouldn't get very far. Rather, one first starts to talk or write about something and at some later stage, one starts to examine the verbiage philosophically. as philosophy of religion, or knowledge, or psychology or whatever. — unenlightened

As disciplines go, it is by far the wordiest, in inverse proportion to its effectiveness perhaps.

If possible, (ideally sic) the best foundation is bedrock, If one has reached bedrock, as Wittgenstein would have it, one has reached the end of that portion of philosophy, the questions are resolved or dissolved, and the superstructure is as sound as it can be.

By contrast, spiritual talk is untethered, lighter than air and floats higher and higher until it reaches such height that it attains outer space, where there is no longer any up or down, and no one can hear you pontificate. — unenlightened

Some philosophers don't know when to stop. A millennium and a half ago the bedrock was sighted and reached in some aspects.

I think of category theory when I read this. It seems to hover over the roughly 30,000 mathematical topics (on Wiki) on angelic wings. A principle component of modern mathematics.

Is there a "modern philosophy", and, if there is, what are its principle components? From Wiki:

Modern philosophy is philosophy developed in the modern era and associated with modernity

:roll:

Great post @unenlightened :up:

I was thinking that the tradition of the "top-level" idea casts philosophy as specifically "the queen of the sciences" — Srap Tasmaner

Refuting Gauss, who termed mathematics as that. Also in medieval terms, theology.

What philosophical notions illuminating the mysteries of QM have been proposed by non-scientist philosophers? Just curious.

Set theory is needed for the rest of math and so is logic — Srap Tasmaner

Much of classical math existed before the introduction of set theory. So, no. Modern math is another thing.

But here I would question whether the notion of cause adds anything that is not already given in the mechanistic description. — SophistiCat

You are correct. It's a mere mathematical simulation of cause and effect. The philosophical notions are out of my league. But it is a fascinating subject.

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

Why should this be the case? I drop an object from a certain height and predict when it will hit the ground. How does this eliminate causality? There are a host of factors involved in this physical feat, and one can argue one's way through that jungle, rather than citing a principle cause, gravity.

(I wrote a math note a year or so ago that partitioned a causal chain temporally so that each link was formed by a collection of contributory causal effects added together to produce one complex number associated with that link. Just a mathematical diversion, but a vacation from the plethora of philosophical commentaries about the subject.)

Which is, again, quite cromulent. — fdrake

Thank you. I just learned a new word. :chin:

But how is that "checking the validity of one argument using another"? — TonesInDeepFreeze

Sorry. Wrong @Hanover post.

I don't know what you mean. Example? — TonesInDeepFreeze

https://en.wikipedia.org/wiki/Proof_by_contradiction — Hanover

Not sure how this fits in with the OP. I've used this approach from time to time, but never to the extent of assuming ~P is true and showing P follows. Usually one shows a logical contradiction of sorts short of P being true. But I digress from the conversation, which has long ago become absurd. @Tones clarified the issue way back imo.

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

I agree somewhat with the "required pace", but you and I know that in post-K12 math and related subjects it takes effort and time to accumulate a background necessary to advance or apply knowledge even a bit. In my years of teaching college math I have encountered only one such individual - an older student who dropped out to support himself as a poker player. He had taken my course in complex variables, and I recall speaking with him informally in the math office in which he brought up a really interesting and unusual notion on the subject, spur of the moment. Like a light bulb burning bright. I was unable to convince him to continue the curriculum.

"the Dismal Science"

But since my big post from 6 days ago was opaque to you I'm quite discouraged from continuing on. — keystone

The one trait I share with Feynman is understanding a concept through examples. You started with the imagery in 2D. Maybe more?

Not sure. But a lot of potential PhDs seem to wash out just from the long dreariness and uncertain job prospects after completion, — BC

Forty to sixty percent of American PhD candidates do not complete their degrees. Many of these encounter a roadblock in the fundamental aspect of the degree: Doing original research. But there are various other reasons. The percentage is a bit higher in the humanities. Lower in STEM disciplines.

Waiting for someone to bring up quantum theory. :roll:

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

The "Lounge" is part of TPF.

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

At first I thought this is not true, but that is because I followed a learning curve that incorporated analytic geometry before calculus, and this allows clear illustrations and examples en route to limits. You work in CS, however, and what seems like unintuitive definitions to me probably make more sense to you. Had I taken a course in graph theory what you are proposing might seem less opaque.

Continue, if you like, and I will comment from time to time as I learn more about graph theory.

I suggest this thread be placed in the Lounge since it obviously has limited appeal to the general audience, but has merit in philosophy of mathematics. Just my opinion.

I think there's also a difference between "memory" or memorizing something and the knowledge (and the understanding) of something. — ssu

Albert Einstein: “If you can't explain it to a six year old, you don't understand it yourself.”

If you can't illustrate a concept or process by applying them to specific examples, you probably don't really understand them.

Richard Feynman: “I don't know what's the matter with people: they don't learn by understanding, they learn by some other way — by rote or something. Their knowledge is so fragile!”

Memorizing without understanding is like sex without orgasms.

I would like to ask the reader about how does the reader suppose that knowledge can influence one's identity? — Shawn

Years ago I completed a curriculum and became a meteorologist and practiced that discipline. Later I completed a curriculum and became a mathematician. I drifted away from the former and lost that identity, while becoming identified with the latter. Meanwhile I became proficient at rock climbing, and that was a separate identity. I married and had a child, so I became a parent. We may have multiple identities.

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

You've lost me. Guess it's time for me to quit. Overall, I think you have started down a path that is far too complicated for the desired result. However, if the result you seek is more philosophical than mathematical you may have something.

It's a philosophical question which most philosophers are not equipped to even begin to answer — flannel jesus

:up:

But what's that got to do with the topic of the thread? — Clearbury

Just checking to see if you are AI.

DKE is accurately characterized as 'the stupider a person is, the less likely they are to realize how stupid they are' — Clearbury

Not necessarily true. I have known stupid people who admit they are stupid and don't try to compete intellectually. But it's not the definition of DKE.

Can you forgive a person for being stupid?

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

Are you trying to demonstrate your argument? Just curious.

Years ago, at a math conference in Marseille-Luminy, a prominent mathematician told me he could forgive someone making a mistake, but could not forgive stupidity. Can you forgive stupidity?

i wouldn't have thought an expert would write a wikipedia page - they're too busy being experts — Clearbury

Mathematicians, specifically, are eager to present their research to the public. Publishing in journal reaches only very select readers. More mistakes occur on easy math pages rather than advanced topics. That's where amateurs demonstrate their DKE.

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

I explore various properties of contours in the complex plane, defining a metric space whose "points" are contours. $z(t)=u(t)+iv(t),\,\,\,0\le t\le 1$.
"Distance" is defined $d\left( {{z}_{1}},{{z}_{2}} \right)=\underset{t}{\mathop{Sup}}\,\left| {{z}_{1}}(t)-{{z}_{2}}(t) \right|$. But here t is a positive real number, which you have not defined yet. Usually, the u(t) and v(t) are differentiable, giving a smooth curve. So incorporating this sort of thing into the definition of vertex assumes what you will probably wish to prove. I wonder what an "edge" in your graph would be?

And isn't Wikipedia written by those who fancy themselves experts in matters they have no expertise on? — Clearbury

It varies. Topics that are popular draw the attention of experts in those areas, who, in turn, make corrections that are discussed on talk pages. Very low interest subjects may exhibit greater DK . The necessity of linking assertions to external sources, books and magazines, e.g., improves accuracy but is not infallible since those references may not be accurate. In general, Wikipedia and Britannica have approximately the same degree of accuracy, but Wiki keeps pace with discoveries faster.

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

It's had 164 years. We'll see. :cool:

Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality — keystone

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.

(Wiki)

OK. But it gets a bit anti-intuitive. Hard to imagine a curve is a vertex. But then, I treat curves in the CP as points in a metric space.

I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn. :roll:

A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, pseudo point, or actual curve — keystone

A vertex represents an actual curve?

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

In fact, I had never heard of the S-B tree before it was introduced on this forum. It is not true that every mathematician will find every math topic interesting. (Wiki lists well over 25,000 if I recall). Had I been a number theorist or a CS person I may have known of it. I see it averages about 47 pageviews per day on Wiki, and classed as low priority. But that's not trivial by any means. My own page gets only 15.

All deterministic processes are time-symmetric — SophistiCat

Not all. The evolution of entropy in a closed system is deterministic (entropy always increases), but it is not time-symmetric because entropy decreases in reverse time.

There are mathematical dynamical systems that function in simple ways that are not reversible. f(z)=z^2.

You could start with continuity described as a path in the Euclidean plane or complex plane taken by a moving particle. Or something similar.

There's an elegance to QM and I believe the same can be said about the top down view of mathematics — keystone

Careful. I would not compare if I were you.

If you had two functions on Q then a suitable metric would be the supremum. — jgill

Suitable for what? — keystone

For defining "distance" between functions. When I dabble in the complex plane contours become points in the metric space and the distance between them is the Sup|f(t)-g(t)| over 0<t<1 for instance.

If I were younger I might have more time to try to unravel your presentation. You have wandered from metric spaces to topology and now graph theory, with that dreadful SB-table trailing along. Then you have all these definitions which a mathematician is unlikely to find of interest.

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

Try a nearby university where a grad student might want a little extra cash.

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

Good luck with that. Probably of more interest to CS people.

The function x(n) — keystone

A sequence of rationals I assume. If you had two functions on Q then a suitable metric would be the supremum.

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

How do you define "continuous"? Are you sure it is indivisible?

Sorry, but your list of definitions is mind-numbing. Your top down is becoming way more complicated that bottom up, IMO. And the irrational numbers have yet to appear.

Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so.