Maintaining a generally equal level of information between all the members of society is of utmost importance . . . — Pfhorrest

I agree in theory. But making the information available doesn't necessarily mean all can or will absorb it. Nature magazine in the 1800s published breakthroughs in science and math, and at that time educated readers could comprehend much if not all that was conveyed. Nowadays specialization has made that virtually impossible. Even being a retired prof of math I can't understand half of what is conveyed to those in my profession. So, how would you overcome the inabilities of the general public to understand specialized information?

freethinking education requires what we might call a proselytizing approach to information distribution: when new information is discovered, that news must somehow become widespread, and not remain only known to those who discovered it and those closest to them. — Pfhorrest

Do you mean new discoveries in math must be promulgated to the general public and not kept locked away in incomprehensible journals for fellow experts?

I admit, when I read your posts I interpret your commentaries from the perspective of mathematics. If I were a pastor I suspect I would see things differently.

This also relates: — Gregory

Thanks. Wasn't aware of that. Kind of anti-set theory regarding natural numbers.

What's the square root of the real number -4?

A car going at constant speed passes point A at stopwatch time=0, then passes point B, one mile further at stopwatch time=one minute. You ask, "What was the speed of the car back there at point A?" Your answer, "It was moving at 60 mph at point A".

Silly, but this entire discussion needs termination.

Say you're a high school student, or an undergrad in college -- you know, general education, like the general public gets already. — Pfhorrest

College algebra is considered under the broad umbrella of "general education" courses. Does the general public get that already? You have a mistaken notion of "general education". I can't recall coming across a course in college called General Education. You appear to be wandering in an academic marsh here.

Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number — TheMadFool

You've got to be kidding. Think about this statement for, say, five seconds. :roll:

↪jgill
They’re not actually waves but the distribution of probabilities. Just so happens they can be visualized as waves. N’est pas? — Wayfarer

:cool:

I don't see how QM indeterminacy can be fitted into mathematics at its foundation — Gregory

It's those damn wave functions! They just seem to be everywhere (at least on this forum). No one mentions path integrals in QT. We can imagine waves, but not functional integrals.

btw I have always been a savage, as a child I never read for pleasure (only out of obligation), didn't like puzzles or board games that were strategic...I would rather do than think — dazed

If you decide not to terminate early, you can carve wooden ducks in old age. Oops, that's artistic. Maybe not. Don't overthink it.

your response is similar to saying that sometimes it could be better if there weren't a separation of powers in government, — Pfhorrest

Not similar, my friend. Your analogy, not mine.

General education for the general public would require testing, and by independent agencies? Sounds very Soviet.

You know, you sound very authoritative in all this, but I don't have the knowledge to engage you, and I don't know if you actually know what you are talking about. But I give you the benefit of the doubt. It would be good if kenosha kid or another real physicists would comment.

The ideal form of such a system of education would, I think, see the pastor role described above as the central figure, to whom laypeople come as students with questions and arguments to be resolved. Those pastors then turn, on the one hand, to the authors of tertiary sources for their knowledge, who in turn turn to authors of secondary sources, who in turn turn to the authors of primary sources; while on the other hand the pastors turn to teachers and to public educators to better inform those laypeople coming to them as students. — Pfhorrest

And this pastor would be . . . . . . . . a philosopher?

A teacher should not be teaching to texts that they wrote themselves, nor testing their own students on how well they have learned what the teacher wanted them to learn; and neither should the one doing the testing be the author of the text against which the students are tested. — Pfhorrest

Maybe in sociology, but in mathematics this is ridiculous. A professor who is a leading expert in a particular area writes a text - mostly his own discoveries in that area - and teaches a graduate seminar in that area. I would say students interested in that area benefit greatly. That's been my fortunate experience. Even as an undergraduate, a professor might teach calculus from his own text to the benefit of his students. I know of instances where this occurred. Of course, calculus has a common core and that puts it in a different category.

Your rules are inflexible. A little like the Volstead Act.

More generally, it is a dependently typed logical programming language, with clause resolution and other rules of logical inference, together with SAT solvers, methods of analytic tableaux and heuristics for automated or interactive theorem proving — sime

Thanks for the information. From my perspective the project sounds dreadful, but for coming generations it may become standard. It's given me a moment to reflect on a theorem I am putting together and proving at present. The intuition and imagination I have used to both design the theorem, then prove it, presumably in the future could be generated in some computerized fashion. But of course I don't see how. :chin:

As for simply formatting the proof of a theorem in a computer language I might be able to do that in a version of BASIC. But why would I?

↪jgill
And you answer? — tim wood

I just brought it up as a topic. fishfry is of course correct. :smile:

The circumference will have digits going to infinity — Gregory

This is indeed a puzzle.

You take a snapshot of a moving car. You look at the photo and ask, "How fast was it going?"

My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind? — GrandMinnow

Professionals, yes. Non-professionals (forum participants, for example) a lot weaker.

today’s generation of mathematics undergraduates who are studying mathematics using theorem provers from the outset. — sime

What's a "theorem prover"? Computer program? A tutor?

If I had more interest in this subject I might do just that. It may be unprecedented but the word "qualia" dampens my enthusiasm. Good luck, though.

Aristotle never understood this stuff — Gregory

I'm thankful for that. :roll:

I'm not looking for people to buy in, I'm looking for truth — Metaphysician Undercover

Time for you to develop a new axiomatic system, then, that leads to "Truth".

So we ought to conclude that "objects" and "processes" are distinct categories. — Metaphysician Undercover

Agreed.

Brains by contrast, with the elaborate quantum machinery of their unique biochemistry, are like quantum suns, radiating entanglement effects on a large scale while overriding classical time dilation. — Enrique

A reference in this regard would be nice. I know very little about the quantum world.

I pretty much agree with everything you have said. I don't see anything novel in what you have written, but you have explained it well. Having worked in academia, however, I have rarely witnessed this degree of perfection. Take reviewing, for instance. I once wrote a paper and stated a simple example in the first or second paragraph demonstrating why a certain result obtained in another paper under slightly different hypotheses could not occur. The reviewer said the paper should be accepted but that the result from the other paper would ensue. I wrote back explaining why that result would not follow. Never heard back, but the paper was quickly published as I had written it. When a reviewer knows and presumably respects the work of an author they may skim over a few details. But this is math, and not biology.

At every stage of the process you have described things can go haywire. However, when the work has important consequences things go more according to your doctrine.

Philosophers don't necessarily lead better lives than others, nor are they more moral, and they most definitely don't know any more about existence than the general public.

In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry

:up:

Sorry. Can't follow what you are trying to say.

Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes? — fdrake

Yes. Goes back a hundred years if my memory serves. Sometimes it's very easy. For example, here is a linear fractional transformation written in terms of fixed points and multiplier.(I'm working on a theorem right now involving this). I think a more general case was dealt with in the discipline of functional equations. Can't recall the work offhand.


$\begin{align}& f(z)=\frac{(\alpha -K \beta )z+\alpha \beta (K -1)}{(1-K )z+(\alpha K -\beta )},\text{ }{{F}_{1}}(z)=f(z),\text{ }{{F}_{n}}(z)=f\left( {{F}_{n-1}}(z) \right) \\ & \Rightarrow {{F}_{n}}(z)=\frac{(\alpha -{{K }^{n}}\beta )z+\alpha \beta ({{K }^{n}}-1)}{(1-{{K }^{n}})z+(\alpha {{K }^{n}}-\beta )} \\ & Define\text{ }{{F}_{t}}(z):=\frac{(\alpha- {{K }^{t}}\beta )z+\alpha \beta ({{K }^{t}}-1)}{(1-{{K }^{t}})z+(\alpha {{K}^{t}}-\beta )},\text{ }t\in R \\ \end{align}$

You dismiss my analysis just because I didn't utilize decimals? — Enrique

Not a decimal person myself. But equations do not determine reality.

If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? — fishfry

The arrow may be momentarily stationary, but it has momentum.

Look up Shota Kojima and infinite compositions. A lot of stuff out there on fractals and simple iteration, but composing different functions endlessly not very much, especially if one considers complex functions that are not holomorphic, which I have done. They are far more interesting IMO.

When teaching almost thirty years I would publish a paper each year just to avoid collapsing intellectually, and all were readily accepted and published, but after I retired in 2000 I decided to do the research for me and anyone else who might be interested, avoiding the formalities of publishing. Hence a collection of informal notes on researchgate. It's amazing the audience one reaches there. Virtually every civilized nation has cropped up with reads of my stuff. But fewer than half read the entire note. Still, that can be 20-30 a week that do. And it's free, unlike most journals - and that irritates me since few journals pay the referees, at least when I was active. (in all fairness, institutions take up the slack with fewer teaching hours, etc.)

I consider the major journals corrupt in their financial practices. But that's just me. Pay no attention.

This notation was suggested by a Japanese mathematician. I was starting to use something else, but switched to his.

What's the meaning of L? — fishfry

Here's what's going on, with a simple example:

$\begin{align}& {{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ & {{g}_{1}}(z)=2z+1,\text{ }{{g}_{2}}(z)={{z}^{2}}-3\text{ }\Rightarrow \text{ }{{g}_{2}}\circ {{g}_{1}}(z)={{\left( 2z+1 \right)}^{2}}-3=4{{z}^{2}}+4z-2 \\ & \underset{k=1}{\overset{2}{\mathop{L}}}\,{{g}_{k}}(z)={{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ \end{align}$

$\underset{k=1}{\overset{\infty }{\mathop{L}}}\,{{g}_{k}}(z)=\underset{n\to \infty }{\mathop{\lim }}\,\underset{k=1}{\overset{n}{\mathop{L}}}\,{{g}_{k}}(z)$

The process arises using techniques from functional equations. For example,

$\begin{align}& f(z)={{e}^{z}}-1\text{ }\Rightarrow \text{ }f(2z)=f(z)\left( f(z)+2 \right) \\ & f(z)=z(z+2)\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2z\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2f(z/2) \\ & \text{ =}\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( 2{{z}^{2}}+4z \right)\circ f(z/4)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( \frac{{{z}^{2}}}{8}+z \right)\circ 4z\circ f(z/4)=\cdots \\ & \text{ =}\underset{k=1}{\overset{n}{\mathop R}}\,\left( \frac{{{z}^{2}}}{{{2}^{k+1}}}+z \right)\circ {{z}_{n}},\text{ }{{z}_{n}}\to z \\ \end{align}$

But here
$\underset{k=1}{\overset{3}{\mathop R}}\,{{f}_{k}}(z)={{f}_{1}}\circ {{f}_{2}}\circ {{f}_{3}}(z)$
etc. Going the other way involves rules for inverses of compositions.

This sort of thing appears in a general study of infinite compositions, a topic of practically no interest in the larger mathematics community. Abstractions and generalizations are more attractive.

So what does the L mean in your equation earlier? Not familiar to me. — fishfry

Give me a few moments. See the Gabriel's thread.

When you consider the predictive power of maths, the fact that through it you can discover things about reality that you otherwise could never know - how is that reconcilable with the idea that it's a 'useful fiction'? — Wayfarer

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

My former and late father-in-law's good friend Eugene Wigner raised this question years ago in a famous paper. I agree - I don't perceive it as a useful fiction.

A quarter counterclockwise turn in the plane. That's the simple meaning — fishfry

That's what happens when multiplying a+bi by i.

I play in the complex plane all the time, and I have always visualized figures and imagery and motion. Even created what might be considered art in the process. :nerd:

Please return to metaphysics. The real deal is a challenge.

Would the fully corroborated evidence of supernatural events necessarily lead one to believe in God, or at least in the supernatural;? — Jacob-B

I'll let you know if it happens. :roll:

The only group that interests me is SU(2). :chin:

↪jgill

That's pretty rad! — norm

One of my inventions (probably! You can't tell in mathematics.) :smile:

hat's been the case in my experience too. For applications, though, the dual numbers are actually important today. Some of the autodifferentiation powering machine learning use the forward method, employing dual numbers to great effect (and even hyperdual numbers.) This allows one to compute f(x) and grad(f(x)) at the same time at low cost. — norm

See, I learned something from your post! Thanks, norm.

but felt that she entirely missed the meaning of complex numbers — fishfry

What would you say the meaning is? Just curious. :cool:

Pi only encodes a finite amount of information — fishfry

Bet you haven't seen this:


$\begin{align}& ArcTan(z)=\underset{k=1}{\overset{\infty }{\mathop L}}\,\frac{2z}{1+\sqrt{1+\tfrac{1}{{{4}^{k}}}{{z}^{2}}}},\text{ }\underset{k=1}{\overset{n}{\mathop L}}\,{{g}_{k}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ } \\ & \pi =4ArcTan\left( 1 \right) \\ \end{align}$

so as a basic dude, the more my body breaks down and my reasons for living become less accessible and then clouded by annoyances like poor sleep, physical limitations and ailments, the balance sheet starts to sway towards I'd rather go out on a high.. — dazed

I've seen this play out in the sport of rock climbing. Sometimes those afflicted follow through. Becoming obsessed with physical performance and not capable of gradually adjusting one's attitude over time is a recipe for termination.

There are times when I think back on my life in the sport, reflect on my performances when young, but I see all that from a different perspective now, and it seems distant and pleasurable but not all that significant. Young, energetic climbers frequently anticipate themselves in old age retaining that fascination that compels them. But it ain't necessarily so. Give yourself a little slack.