Not a finite set of axioms, rather a countably infinite set of axioms. — TonesInDeepFreeze

I hoped you would chime in. :cool:

But Godel was speaking of a small finite collection of axioms, not an axiomatic system that continues to increase without end. At what point does one initiate the drawing of conclusions? Tacking on the axiom of choice took math into new dimensions, as did infinity axioms. BOAK seems bewildering rather than enlightening, imo.

Everyone desires some advantage, some way to be better, smarter, faster, stronger, more talented, more charming or more beautiful than others of of our species. But we're not all willing to pay the same price or make the same amount of effort or take the same risks to achieve it — Vera Mont

:up: :up:

But there is a way to deal with the inequalities and be peaceful & honest. — YiRu Li

The Meek shall inherit the Earth

Undecidability cannot possibly occur. — PL Olcott

Not undecidability. Rather potential facts. I know my Corgi could not understand analytic geometry, and that is a general assumption for Corgis. But there is the faint possibility that one will come along and understand the math. Then that fact becomes a member of the set of "axioms" in your system. But it is not at present. Thus your system continues to grow, and with each new axiom there is the possibility of contradicting a previously established axiom. So, when you ponder Godel what system are you talking about?

My question remains: show how exactly all axioms can be listed for reference. What is axiom #1?, #2?, . . .

The Cyc project has {Thing} at the root of its knowledge tree — PL Olcott

The objective of the Cyc project was to codify, in machine-usable form, the millions of pieces of knowledge that compose human common sense.

A far cry from listing all facts (axioms).

The body of all current analytical general knowledge is not only countable it is finite. — PL Olcott

So, the system of axioms is constantly increasing. Proof it is finite at a particular time?

What I mean by axiom is any expression of language that has been stipulated to be true — PL Olcott

"True" by what measures? What of potential inferences not realized?

Season's Greetings

Merry Xmas, Lads.

But the space itself is expanding within any duration of time — universeness

My understanding is that two objects move further apart with time; space itself (whatever it is) doesn't change. Some of this stuff is assumed axiomatically.

All of the basic facts of the model of the current world are stipulated to be necessarily true, thus are the axioms of BOAK. — PL Olcott

OK, this means an uncountable collection of "axioms". How could you organize these axioms in such a fashion they represent a data set in CS? What is axiom #1 ?

Some time back we had a promising theory of everything that started with the premise all facts could be catalogued within a program. But when asked "how?", things began to fade.

and we should accept that in mathematical terms, the distance between A and B cannot be traversed unless 'infinite acceleration' is a real thing? — universeness

:lol:

But, in Minkowski spacetime it seems progression in the time variable requires no movement in space. I think of time as being in some ways linked to movement - a philosophical perspective. Just a feeling. JS might be able to address this issue, which is probably a triviality on my part.

There was another quantum physics guy on the forum a couple of years ago, @Kenosha Kid. He said he left physics to make his fortune playing the guitar. Well, actually he chuckled at the "fortune" bit. He was a transactionist.

In my naive fantasies I wonder if our macro world is indeed a simulation wherein the creative mechanisms are intentionally hidden from us in the quantum realms. With math we are able to manipulate results down there to some extent but remain puzzled at non-locality - a feature of that realm where scales of measurements and dimensions might be easily manipulated by our overlords. :roll:

In relativity theory it seems there is possible the passage of time without any physical change. Time being completely independent of change seems peculiar. This, unfortunately, puts me somewhat in MU's camp: we don't truly understand either time or space. But I am old and have lost brain cells. :chin:

, ,

As I sit on the sidelines I find this conversation fascinating, and I am learning some math as well. I've never worked with formal integral transforms and am surprised at how many there are. I now understand a bit more about the uncertainty principle.

(IMO Zeno should be dead and buried)

My theory about his attraction to dictators and facists is that it’s not grounded in political theory, but the simple fact that they wield the kind of power that he craves — Wayfarer

I agree. But his following continues to grow in a world turned upside down.

Also, I only just learned that this environment is tex-enabled too, so that's nice. Can I ask how you did that? I can't seem to figure it out. — Jaded Scholar

I use MathType, what you see is what you get. Then copy and paste to this forum, changing "<" to "[" and so on. But there is a tutorial here on MathJax I think. Enter the word in the search box.

Thank you for your comments about the use of series in physics. A power series is one kind of infinite composition of complex functions. I got started in the subject with another, continued fractions. Then there are infinite product expansions. Finally, examples I came up with after a colleague pointed the direction, elementary functions

The Bogoliubov transformation is a new one for me. Not quite sure how to decipher it in Wikipedia. Some symbols probably physics related.

When I got my degree fifty years ago, a professor told us that we would never know as much math as we did then. He was correct. As the years flow by we mathematicians get more and more entrenched in our specialty, and don't get off that path very much, allowing progress in math to surge past us. At my age I'm lucky to remember elementary topology or group theory. :worry:

Mathematics is commonly classified by philosophers as a form of art — Metaphysician Undercover

You and I have our differences, but here I am more or less in agreement. Mathematics is a practice, a device used by the sciences, etc., but mathematicians would largely agree it is a kind of art, requiring imaginative progress - the canvas upon which I apply my mental brush is the complex plane.

As for failure in mathematics, le mieux est l'ennemi du bien.

I am a leftie democratic socialist Mr Gill, I hope that does not lower your opinion of me too much. — universeness

Wife & I are still registered Democrats. You might be surprised at where we would agree on politics. For example, I have always felt there should be free education all the way to professional degrees and PhDs, and there should be free health care for all. I firmly support Medicare and Social Security, along with defined benefit retirement plans. The USAF made me a meteorologist at the U of Chicago, free of charge and fully financially supported, then, later, the GI Bill helped with my terminal degree. :cool:

But our country's immigration catastrophe and a few other issues, like overly liberal law and order policies, pull us to the moderate right at times.

OK. So much for all that. My opinion of you is very, very high, Buddy ! :smile:

This is huge — Wayfarer

It will be overturned by the Supreme Court. I know you think it is justified for this person, but even some Democratic strategists are calling it a step too far. A Democracy in which one cannot vote for a popular candidate of a major party because the reigning opposing party won't even allow his name to be penciled in on the ballot?

Hey, I love Bret Baier ! :cool: — jgill

I just learned who that is and I hope this is fake news. :joke: — Jaded Scholar

My wife and I usually vote moderate conservative these days, and we both enjoy the evening news with Bret. A tilt to the right, but maybe less than CNN's tilt to the left. The Fox commentators is another matter. Hannity and his ilk are way off my radar.

It's a program/programming language that is basically C, but combined with some Java to build higher-l — Jaded Scholar

I meant the complex plane, C. Not a programming language. And I'm actually talking about

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

And going the other way, as well, in physics (mathematical) research? A contour in C looks like $z(t)=u(t)+iv(t)$ , t = time.

Oh, man. If Trump would just employ the mechanism most of us have that mediates between what pops into consciousness and what comes out of our mouths . . .

What a choice upcoming. I cannot forgive Biden for ignoring the border crises.

I'm glad that the only thing I've ever experienced on this front is when physicists semi-jokingly check the room for mathematicians before writing (Δx)² ≈ 0 or sin(θ) ≈ θ for θ<<1. — Jaded Scholar

Hey, I use that. The wiggle makes it true. :smile:

Your discussion with Uni is the best thing I have seen on this site.

turn this site into something akin to fox/fake news — universeness

Hey, I love Bret Baier ! :cool:

Just off the top of your head, can you think of instances where complex functions are composed? I appreciate your comments about string theory and spin in that subject, in particular. I'm looking into compositions of contours in C and I have wondered about compositions of strings.

Physical impossibility is admittedly just a possibility for us; we are epistemologically limited, so we don't know with certainty whether anything is physically impossible or not — Janus

We are limited by what we do not know that which we do not know. What is beyond our realm of perception is to some extent beyond our realm of conception. As my late algebra student, my Corgi, Jake, would attest.

It seems professionals in science or math become uncomfortable in this environment. My own experience is my first thread about the diagonal paradox, to which there were some negative replies going beyond mere analytical comments to what seemed personal attacks. But, at least in those days science or math topics were not relocated to the Lounge.

At that time there were a couple of grad level math people here, but after a while they experienced the fatigue of constantly arguing with members who were minimally acquainted with the subject, but held strong views against accepted perspectives. When the competent are belittled by the unknowledgeable the former tend to move on to other conversational pastures.

I'm so old I don't really care. When MU pops up with his highly literate arguments about mathematics I indulge and appreciate his perspective if I can. But I also know that if I open my TPF mouth about QT I wander into unknown territory and my thoughts are trivial.

If those who wished to argue science were Gentlemen, like you my friend, or Ladies, some of those who left might have stayed. But this is all conjecture. And philosophy is all about argument.

The mathematics supporting quantum mechanics is entirely logical. But, although there are various metaphysical interpretations of QM, are there metaphysical analysis of this particular math structure?

The metaphysics of mathematics seems to be the existence of a mathematical object in a mathematical structure. Not the structure itself.

If quantum theory becomes philosophically stable at some time in the future all this stuff that spins off its mysteries will probably become scientific ectoplasm.

I just heard a pin drop. :yawn:

Is risk vs reward a proper philosophical subject? I don't see much here to indicate it is. The world is full of stories of those who have succeeded in risky ventures. We learn from these tales, and they may help us as we toil away at life. But, is there a philosophy of risk taking?

What did Aristotle have to say?

Hunter is becoming legendary. An Errol Flynn for our political age, sailing the American Main. Like Trump, the more he is pummeled the more he is admired. Pay attention, GOP !!

Yes. Impeachment effort seems ridiculous. Especially since Trump leads in the polls.

Prepare yourselves for a second Trump term as President. Borderless Biden has been asleep at the switch too long. Financial crook or senile elder?

There is no Purpose Fairy flying around at night, dispensing reasons to live. You have to find your own. — Vera Mont

:up:

Try to come to grips with existentialism. There must be something that gives you pleasure. Discover paths that lead from that activity, even if it is an activity hardly worth mentioning to others. Use your imagination to illuminate and create value with obscure discoveries. What is tiny can become big.

The interplay of different kinds of risk you report sounds scary. My inner OSHA supervisor is trying to steer me in the other direction. — Paine

Alex Honnold starred in the Oscar-winning film "Free Solo" and became the world's most famous rock climber. He epitomizes a certain justification of controlled risk in the climbing community, although he lost several sponsorships in the process. His spectacular accomplishment was supported by considerable practice and an innate ability to confine emotion to allow near physical perfection - a degree of calmness not seen often in that community or society in general.

I propose "curmudgeonlogy". — wonderer1

Finally, A breakthrough on TPF, an original philosophical path that must have the amateurs and professionals here salivating! I can imagine dust blown off ancient texts searching for what Aristotle had to say on the subject - if anything at all. :cool:

Intervention by means of a mean old person . . . — Paine

the roster of mean old people — Paine

What's with this categorization? Is there a name for a philosophical study of "mean old people"?

The philosophy of risk should be the psychology of risk. As a former rock climber for over fifty years I have observed the interplay between physical risk and reputational risk.

but you do wonder why they don't incorporate a 'maths module' based on something like Mathematica, — Wayfarer

This should work for definitions and examples and evaluations of certain complicated computations. As to reasoning and the guess work that goes into problem solving I'm a little pessimistic. How do you program creativity? I am currently looking into compositions of contours in the complex plane, and finding ways to compose that violate standard usage of the term as well as more conventional approaches. Mathematica is a machinery for what I come up with. Just a trivial example.

There's no middle ground between existing and not existing. — RogueAI

There's the problem, whether regarding the moon or a quantum particle. Were philosophical terms defined more clearly some threads would vanish. Exist physically, or exist metaphysically? One or the other or both or neither? Then there are those damned probability waves, always collapsing like snowcones on a summer day just because we stare at them.

The moon shows a tendency to persist when we look away.

So mathematically, you can create an nth dimensional array, and such an array would exist in reality, but cannot be geometrically displayed in 3D — universeness

Done all the time: n-dimensional vector spaces. I think the Hilbert space ( a special kind of vector space) in quantum mechanics may be infinite dimensional. Then some kind of linear operators are defined on it.

If pushed, would a mathematician be willing to say something such as 'well you could think of the 'imaginary number line,' as in a sense, 'wrapped around' every coordinate in a standard 3D coordinate system, such as (x,y,z), or (x,y,z,t), t being time — universeness

Sure. There are no limits to how bizarre math can become. This doesn't seem so outlandish.

Some physicists, like Smolin would say that string theory is done — Metaphysician Undercover

:up:

Spacetime, therefore, is not all there is to reality. — Wayfarer

Would you want this on your tombstone? :smile:

No matter matter who, the important difference is Unlike topological dimensions, the fractal index can take non-integer values, indicating that a set fills its space qualitatively and quantitatively …‘

-Fool — Bella fekete

I'm not following you. But that's OK, I am not a philosopher.

Here is what I think is a very good way to look at these two planes — Metaphysician Undercover

An entertaining and fanciful philosophical abstraction of a bit of mathematics. But we are looking at two lines, not planes. The real axis and the imaginary axis - producing the complex plane. :cool:

This comparison is actually grounded on an inverse topological layout — Bella fekete

I take it you use "topological" loosely, like some use "fractal" - both well-defined mathematical terms.

That series of conjectures is more convincing than not even by the progression within this here forum, of the con-foundation of intended disposition — Bella fekete

??

I don't know if you can accurately say that is "the world". Isn't it more like two distinct perpendicular worlds, the world of real numbers and the world of imaginary numbers? — Metaphysician Undercover

It's a world in which a being embodies the characteristics of two genres, the rational and the imaginative. A bit like mankind.

Thanks for the links. Particularly the one concerning a reduction of ST dimensions to two at quantum scales.( I continue to dabble in the complex plane where the world is two dimensional.) :cool:

I must say I find most of this fairly much incomprehensible — AmadeusD

We need not objectify the self so that we share definition completely. — Vaskane

A clue