feeling of control and freedom, and continuous physical flow. — jgill

:rofl: — TheMadFool

Well, you just roll around on the floor, missy! :razz: I was at university with this fellow:Mihaly C., and he went on to elaborate on the experience, expanding it far beyond climbing.

What does any of this have to do with flat Earth?

Many possible reasons of course but one that interests me is it amounts to negating one's life and survival instinct which is, in a way, rejecting your sense of self-awareness. — TheMadFool

There may be a whole spectrum of attitudes and reasons for extreme sports. Certainly, there is the exhilaration of substantial risk in some of them. I was a rock climber who did a huge amount of freesoloing (without equipment, etc. - you can look it up) so the risk was there but what was most enticing was the feeling of control and freedom, and continuous physical flow. Learning one's limitations and operating accordingly. A kind of dance on the rock.

This is obviously not what you are describing. Perhaps there are others here who have engaged in activities where "negating one's life . . ." They might add an interesting dimension to this discussion.

I would be surprised if anyone speaks up, though.

. . . extreme sports enthusiasts essentially downplaying the value of life (death's ok). In one case you turn off metacognition (lose self-awareness) to make death less painful and in the other case you devalue metacognition by taking wild risks. — TheMadFool

It makes me a bit uneasy when I read this sort of thing about extreme sports. It's similar to the Simone Biles' thread where everyone except a gymnast has something to say about her actions within the sport. But I suppose this forum is designed for somewhat naive discussions of almost any topic. As a participant in both gymnastics (ten years, long ago) and an extreme sport (fifty years) I see things differently. It's more complicated than you might think. If you are a participant you are on a different wavelength.

Wingsuit BASE jumping leaves me speechless, however. :fear: A sport I would never have tried!

I think that the most responsible thing to do, though it is kind of demoralizing for me to say so, is to begin to prepare for the refugee crisis, which, I am sure, has already begun. — thewonder

Will the Taliban allow refugees to flee the country once it is in power?

A circle and a square are topologically equivalent. In a sense they are the same collection of points, deformed continuously from one to the other. So, in a sense there is a shape-changing entity having many forms. Well, that's a philosophical way of looking at it. :roll:

From a retired mathematician who still dabbles with it, when I work on convergence theory in a dynamical system in the complex plane I always demonstrate theoretical results with computer imagery examples. I am doing that at present, and it is gratifying to watch the sequence of dots approach a fixed point as predicted. That sequence of electronic dots has a kind of "physical" existence but is still in a way non-physical. How does this fit into the current discussion? :cool:

You make it sound attractive, but democracies are messy and making and moving on decisions can be ordeals. Look at the problems with the infrastructure bills. But I'm naive on the subject.

Eudoxus (370 BC) laid the foundations for integral calculus by approximating areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. Occasionally a calculus course is taught from this historical perspective, with integration first, then differentiation.

In mathematics an example of a two dimensional manifold is a sphere. Each point on the sphere is approximated by a flat plane (very small). As we stand on Mother Earth we are on one of these planes.

Therefore, the Earth is flat! :wink:

Lots of armchair athletes here — jgill

We are so strong as athletes, that we can lift the very armchair we sit on. — god must be atheist

Being full of hot air makes that possible. Be sure to tether yourself to the living room floor. :cool:

3. At age 15 I encountered dy over dx (which the ratty teacher irrationally insisted, doesn't cancel to y over x) and a snake shape. It wasn't admitted that these had any purpose, nor was the "procedure" explained. — Fine Doubter

If I had my druthers I would have high school math teachers end with a decent introductory course in analytic geometry, rather than do what they might do to calculus.

Recently published material I encounter has the advantage of reverting to the ancient approximation goal, but the continued disadvantage of keeping their abstract method impenetrable — Fine Doubter

Would you expand on this or provide links?

From CNN, Elle Reeve:

She said in morning practice that she had a little bit of the twisties. The twisties are a mysterious phenomenon -- suddenly a gymnast is no longer able to do a twisting skill she's done thousands of times before. Your body just won't cooperate, your brain loses track of where you are in the air. You find out where the ground is when you slam into it.

Coming back to Simone: he or she (I follow sports so little I don't even know his or her gender, race, or citizenship) is mentally ill — god must be atheist

When I get up from a chair too quickly I experience a moment of wooziness. I suppose I am mentally ill. Or maybe I have a mental disease.

Abstract models often seem like 'babble' to people who prefer a more direct and practical approach. — hope

Reasonable abstract models would be welcome. Comparing Simone's "twisty" problem with the Blitz in England during WWII? (I suspect this was satire). And "mental health" doesn't mean she needs psychiatric treatment - it's a specific glitch in a dangerous and very complicated performance.

This is not John Wayne territory.

From Wikipedia:

Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives.

Where does physics end and math begin? Lots of unknowns here.

An entire thread of clueless babble. Lots of armchair athletes here. :roll:

It's not "mental health" the way you may think. I was a gymnast many, many years ago and recall how little glitches can occur in complicated routines. Especially when twisting and turning at high speed. If you have a near miss it might affect you for a while. Once you start thinking about the experience things can get worse. You can't think. It's all programmed.

It's so much more complicated than running around a track or swimming in a pool.

My understanding of quantum mechanics is its not an observer that causes outcomes, its active measurement — Philosophim

Me too. Good post.

Philosophy costs nothing. And is "worth" nothing. Sad but true.

Ditto for lots of mathematics. :sad:

In this context, infinite summation is defined only for converging sequences — TonesInDeepFreeze

(converging sequences of partial sums: S(n)= a(1)+a(2)+a(3)+...+a(n) -> S, or a(n)->0 fast enough)

It's nit-picking, but there are several "summability" theories of divergent series. They assign "sums" to certain divergent series and must give the proper sum to convergent series.

Summability of Series

And Riemann showed that a conditionally convergent series can be rearranged to "sum" to any value.

Update: he retired from teaching in 2019. Look: https://www.coursicle.com/lavc/professors/Harold+Ravitch/
So, for more than 50 years he was a teacher. Probably he has a lot of papers related to this — javi2541997

I can find only two papers, neither related to this topic. If he taught at a community college it might have been difficult to do research and publish.

The question then is, is math discovered or invented? — TheMadFool

Most math people are not concerned with this question. I believe as new math appears on the scene it may have elements of both.

I'll come out straight up and say that my own prejudice is that maths is made up as we go along, but I will admit that I do not have sufficient background to argue saliently for this position. I doubt anyone in this forum does — Banno

Some of it is. For example I recently defined an "attractor form" for a certain class of functions. It's trivial stuff, but may not have been around before. Once defined, then its characteristics are explored. It's hard to tell. My advisor fifty years ago went so far as to state, "There is nothing new in mathematics." Who knows?

A moderator should speak up and explain.

Like a mathematics department, nlab as an encyclopedia is obviously going to disseminate mathematics in a politically neutral fashion. Or perhaps i should have said, unlike a mathematics department — sime

Campus or departmental politics regarding subject matter and its relative "importance" can depend upon current interests of the faculty - and grant money. Sometimes a group of faculty convince administration to focus on or emphasize a particular topic those mathematicians are eager to pursue. Since math is a social activity directions may be decided by social interaction - and strong personalities. And, not least among the reasons for these choices, a department may decide to build around a well-known and accomplished colleague. The awarding of grant money is another story, similar to the above.

Category theory seems topical these days. My old clique - largely kerpunkt - hoped for a resurgence of interest in analytic continued fraction theory. The leaders have simply passed on. RIP

I believe that is a rule of logic, but, yes, I'm thinking more of addition. — Antony Nickles

Yes, of course. But even the notion of addition was expanded in 1801 when Gauss introduced the modern concept of modular arithmetic. As formalized, this was a new idea that could be considered a rule in certain circumstances I suppose. Perhaps only a "rule of thumb." Sorry to interrupt your conversation.

The structure of the rules of math makes them determined in advance, encompassing all applications, eternal — Antony Nickles

Depends on what you refer to as "rules of math". For instance, the Law of the Excluded Middle is useful in traditional or standard math, but not allowed in constructive math. Turmoil in the jungles of the mind.

Interesting to see what PhD theses in philosophy look like. The ongoing search for originality.

if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. — jgill

Nice to have an actual mathematician around here! — fishfry

But one left on the ground while math, like Buzz Lightyear, has gone "To infinity and beyond ! "

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem. — fishfry


Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, but i see no counter-intuitive examples in what you present. In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable. — sime

If one doesn't wander into transfinite math, Choice is not necessarily required. A finite dimensional vector space doesn't require it for a basis. And if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. However, much modern math goes beyond these restrictions and requires transfinite results.

. . . which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis — sime

When I write mathematical analysis programs for my computer I am certainly involved in constructive math, but pen on paper, not necessarily. I suspect constructive analysis will not overwhelm the math community, unless you know something I don't - which is possible. :cool:

Take heart, MU. You may think yourself alone and ridiculed, but you have only to find the Yellow brick Road and follow it to Emerald City where your ideas will find acceptance. :nerd:

In mathematics there are "wandering sets" where points wander off under iteration. The same can be said of threads in which someone goes off on a tangent and gains a following, never to return to the original topic. Is it even possible to force adherence to a topic? Is it even wise to insist?

I felt he [Andy Clark] was re-inventing the Vygotskisn wheel. But I also supported him in bringing the constructionist model to a wider audience - the mind science crowd. — apokrisis

Out of curiosity, did you know my old high school friend Zach Hall? He created a department of neurobiology at UCSF in the 1970s.

Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical. — Metaphysician Undercover

Here's a counterpart of this idea in mathematics. (1) is called left composition or outer composition, and (2) is called right composition or inner composition. (1) and (2) are very different ideas. But to compound difficulties in language or notation, (2) is usually numerically evaluated using backward recursion, which is very efficient:

(1) ${{G}_{n}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ }G(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{G}_{n}}(z)$

(2) ${{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n-1}}\circ {{f}_{n}}(z),\text{ }F(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{F}_{n}}(z)$

And this is exactly my position in a nutshell — kudos

Good! Glad to hear. :cool:

I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. — fishfry

I agree. It's nonsense. Regress means going backward. I am more than familiar with these notions, as I investigate dynamical processes going forward as well as those going backward.

"Lastly, logical regress does not apply in mathematics" - OK, thank goodness.

High praise, seeing as you would have a better grasp of what it is about than we commoners. — Banno

Fishfry and Tones have far better grasps than me of the logic and set theoretic aspects of this subject. I just like the observation that naive set theory has some value. :smile:

This paraphrased excerpt is from a book called 'Calculus and Analytic Geometry by George F. Simmons. — kudos

The late George Simmons taught at Colorado College, not far from where I live. He is a marvelous author, and his Introduction to Topology and Modern Analysis is my favorite math book. The material you quote shows how well he conveys ideas to the reader, going from a formal statement of a theorem to examples of great clarity illustrating that theorem. This is the way mathematics is taught.

What you call computing seems to be simply looking at examples - frequently involving numerical calculations - that give students a more comfortable framework for understanding concepts. Examples of this sort go hand in hand with formal theory, making the latter more palatable. There is no conflict. No pitting one approach against the other. Examples and graphical interpretations are part of heuristics. There is no "opposition".

I know nothing of Hegel and what he thought in this regard.

When I taught a senior level course in complex variables I would try to give the motivation behind theory, avoiding the strict formal proofs in favor of simplified approaches that might appeal to intuition. Now, a graduate level course in that subject is more sophisticated and entails stricter arguments. Even there it's possible to spark intuition.

If the moment of conceptualization takes place through difference and identity, mathematic description and analysis can not take the place of computation — kudos

Now, where were we? :chin:

If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is deducible from logic via the naive comprehension schema.

At first I thought these ideas nonsense, but upon further reading, noting that nothing done here likely disturbs the course of mathematics as it is normally practiced - outside of foundations - I find these notions appealing. Of course, having been exposed to naive set theory many years ago and thinking it the real deal, only to be shocked by what has overtaken it, I am a tad biased. :cool:

↪jgill By computation do you mean reasoning?

Yes, I do mean that in a certain sense too. Do you suggest it is an example of reason and no computation? — kudos

In doing math research, one frequently uses previous results (theorems or axioms) to verify steps in logical arguments. And sometimes actual numerical computation is required as well. But it may be that your definition of computation means use of a computer in a larger sense than merely number crunching. Euclid's theorem you presented is a logical argument with no computation I can discern.

In this proof Euclid employs some computation of a number of prior theorems — kudos

By computation do you mean reasoning?