RIP, Agent Smith. Your enthusiasm and wit will be missed.

After reading the Wiki piece it seems to be more a decision theory problem, and that's how they classify it. On that same page there is the following similar problem that for me is easier to understand:

Sailor's Child problem
The Sailor's Child problem, introduced by Radford M. Neal, is somewhat similar. It involves a sailor who regularly sails between ports. In one port there is a woman who wants to have a child with him, across the sea there is another woman who also wants to have a child with him. The sailor cannot decide if he will have one or two children, so he will leave it up to a coin toss. If Heads, he will have one child, and if Tails, two children. But if the coin lands on Heads, which woman would have his child? He would decide this by looking at The Sailor's Guide to Ports and the woman in the port that appears first would be the woman that he has a child with. You are his child. You do not have a copy of The Sailor's Guide to Ports. What is the probability that you are his only child, thus the coin landed on Heads (assume a fair coin)?

Those results, in my uneducated view, are pretty devastating for mathematics as we know it. Philosophers are probably more inclined to take his theorem seriously. — Ludwig V

In fact, a large majority of mathematicians go their merry way, ignoring the incompleteness thing. Although it's in the background - like being struck by a meteor while playing golf - it is inconsequential to most research. I think you are correct in that it is a more important concept for analytic (set theory/foundations) philosophers.

The thought of being wrong in an interesting way has a charming appeal — jgill

It certainly stuck in my mind — Ludwig V

Years ago when I was still somewhat active in the research community I published a paper on an unexplored topic. I proved a theorem, but with heavy hypotheses. A year or so later a colleague published a similar paper, based on mine, and proved a much, much nicer theorem - minimal hypotheses. So, in a sense, I was "wrong" to assume more than was needed, but wrong in an interesting way.

A professor well known for his contributions to logic once confided in me that he understood Gödel's famous argument, but didn't believe it. (!) That's a consolation for people like me who find logic very difficult. — Ludwig V

My first grad course in math back in 1962 was an introduction to foundations and set theory. It was pretty neat how we (doing homework exercises) started with the empty set and ended the course with a definition of the exponential function. But beyond that, only perhaps two students out of a class of maybe 25 found the remainder of the subjects attractive. Even the young and enthusiastic professor recommended most of us stay away from the topics in the future.

Gödel's results are reflected in only a very small number of research themes.

Haley will be Trump's running mate. I know. Sad times for moderate Republicans. :sad:

Thanks

Sleeping Beauty Problem

While we're talking about it, got any idea what a 4D paradox looks like? — ucarr

Thing is I don’t understand much of what you are claiming — Banno

I'm lost here, too. :roll:

utilizing the ascending sequence of dimensional complexes as steps that collectively establish said consistency. — ucarr

What's that?

If math exists all math exists potentially — EnPassant

I think the Platonic realm does exist in the sense that it makes all kinds of math possible but not necessarily realized — EnPassant

This is an excellent point. Once a concept is defined within an existing framework of mathematics, in a sense all that logically flows from it is potential, awakened by diligent investigations and discovery. The question remains, When a new concept seems to appear out of nowhere, is that creation or discovery?

(I speak from personal experience, not philosophical conjecture)

Sean Carol is a realist about the wave-function — Marchesk

In quantum physics, a wave function is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it.

Wikipedia

I can't see how this (a wave function) could be construed to be a physical reality, but some do. Perhaps my short-comings. :roll:

What are dimensions doing in set theory? — Banno

A vector space is a set of "objects" whose "dimension" is the cardinality of its basis. But this is linear algebra rather than purer set theory. You made a good point. :up:

What would make a fractal finite? And secondly is a fractal really a true fractal if its pattern ever comes to a stop, if the scale of repetition is ever limited? — Benj96

Mathematically, a fractal (that is not finite) is generated by iterating a rather simple function in the complex plane an infinite number of times. So, a "finite" fractal would result from iterating a finite number of times. Then it's not "turtles all the way down".

In common usage, fractal means just repetitive patterns at various scales. Ideally, all scales.

So once inflation ends, the multiverse begins, until De Sitter space, when there's nothing left to decohere and make observations. Then all is just superposition. — Marchesk

Physicists have coined words and expressions as conveniences as they work through the math, and we in the lay community have adopted them as if they really mean something. "Superposition" gets way overused, and frequently in a kind of mystical fashion. "Curved space" is right up there, too.

Once these words and expressions get into common usage philosophers begin using them as they explore metaphysical realms. It's quite a game. :chin:

I can't get it to open. Sorry

Perhaps you could write it out so those of us not wishing to sit through a video could read it and reply.

My perception of the nature of philosophy keeps changing. — jgill

I'm not clear whether you think that's a bad thing or a good thing — Ludwig V

As a math person (retired) sitting on the sidelines, I am beginning to appreciate the dialectics, the vitality of the practice. The thought of being wrong in an interesting way has a charming appeal. :smile:

...disagreement is what keeps us going. — Ludwig V

I can't resist the urge to reply "No it isn't!" — Banno

:rofl:

It looks like the anatomical structure of the brainstem. — Benj96

You might find the images in this entertaining: A Space of Semi-coupled Contour Integrals

Fractal attractors are a common feature of complex systems, so most likely fractals do represent a significant feature of the universe — Pantagruel

I wasn't familiar with "fractal attractors", and found only this paper using that terminology. I suspect what you mean is "strange attractors", which have been studied extensively. But thanks for piquing my curiosity. :cool:

Answers are not the point, and in fact are the death of philosophy. Similarly, agreement about the answers are welcome as an episode, but disagreement is what keeps us going — Ludwig V

My perception of the nature of philosophy keeps changing.

The particular, eternally persisting, elementary physical stuff of the world, according to the standard presentations of relativistic quantum field theories, consists (unsurprisingly) of relativistic quantum fields

The wisdom of the ancients simply turns a leaf and emerges in the now. Nice commentary, though the NYT paywall is annoying.

What do you think jgill? — L'éléphant

One of the reasons I stay on this forum is that I learn things about math. Fractals, for me mostly means fractal art. Fractal analysis of phenomena it seems uses approximations to repeated patterns in skilful ways. From Wikipedia:

Unlike theoretical fractal curves which can be easily measured and the underlying mathematical properties calculated; natural systems are sources of heterogeneity and generate complex space-time structures that may only demonstrate partial self-similarity.[17][18][19] Using fractal analysis, it is possible to analyze and recognize when features of complex ecological systems are altered since fractals are able to characterize the natural complexity in such systems.[20] Thus, fractal analysis can help to quantify patterns in nature and to identify deviations from these natural sequences.

What has happened to fractals is similar to what happens to interesting concepts in math: everyone takes off in all kinds of directions with it. I haven't followed these developments with fractal theory because they didn't interest me much. I've been more concerned with what happens when infinite sequences of functions are composed. It's virtually all nonlinear in the mathematical sense.

Thanks for bringing this to my attention.

If you have a computer assisted proof, do you know the result? Suppose it is a proof that you cannot follow; is the feeling of certitude necessary for the claim that you know the answer? — Banno

The Four Color theorem was an interesting start in this direction. Some professionals still have doubts, while most reluctantly accept what the computer has done as "fact". I write BASIC math programs frequently to guide my intuitions, but I hope to never use a program to actually "prove" something. Luddite me.

It all becomes a matter of trust, and I fear the profession will ultimately call yield and allow CS to do the hard work. Then mathematicians will be relegated to philosophical discussions trying to interpret what the program has come up with. :worry:

The universe is not a fractal. There are no rules — Benj96

I don't think the universe is fractal, but I do think there are some rules. "Fractal" began as a mathematical notion that arises from simple iteration of a single function ("rule") in the complex plane. But as time progresses you may have noticed things change. Why not such rules? Instead of a single function perhaps there should be an infinite sequence of functions that are iterated, one after the other. (Confession: this is my mathematical area of research).

Fractals and chaos and everything in between can result from simple iteration of a single function. But when you use a sequence of functions you might be surprised at what comes forth. Look at my icon on TPF.

I thought the same... but it looks like that we are convincing Agent Smith to think otherwise! — javi2541997

Yes, he is capable of doing a merry dance amongst the partygoers here. :cool:

Medieval scholasticism? :roll:

Is Arche more akin to "first causes" or axioms or postulates from which first causes might emanate?

the Department had to attract more students, and so was to both accept students with less ability and offer less demanding courses — Banno

We (mathematics) were mostly a service department, with courses we designed for liberal art majors, business majors, engineering tech majors, etc. We occasionally had run-ins with those departments about grades and standards, and we could adjust those without abandoning our self-respect. I mostly became responsibly involved in these things when I headed up the department for a couple of years.

Because of the service factor we were able to keep courses for our majors at a reasonably high level, with me designing and teaching the senior level offerings in real analysis (intro), complex variables, and topology. I was considered the most demanding, but mostly about giving a B instead of an A. :cool:

The result is apparent in this forum. Folk think philosophy easy, a topic for dabbling dilettanti — Banno

I plead guilty. I am learning slowly about what makes a philosopher tick. My one senior level course in the subject in 1958 was more of a survey and little was said about the practice of philosophy.

It's just preferable to argue about the meaning of "gavagai" on a full belly. That's pretty much the reason i decided not to pursue academia — Banno

You must have had a feeling of knowing what was in store. Or was it more than that? :smile:

Seems to me that there is a play on two senses of "know" going on here — Banno

And the distinction between "knowing" and the feeling of knowing. When I follow the proof of a theorem I know the theorem is true according to the rules of the game, and I have a feeling of knowing. But when I park my car out of sight I only feel that I know where it is. So it is conditional knowledge. Most of life is lived in a complex of probabilities.

But then I have the feeling of knowing what I have written is of no consequence in this forum. :roll:

Interesting link. It's not quite what I had in mind, but probably more reasonable. I had thought of a continuum between the world of ideas and the physical world.



Thought-provoking. Thanks. Goes back to constituting mathematical (and physical) reality from the continuum to the discrete, not the other way around. Traces of Bergson.

Mathematics, it seems to me, is like them in that respect - it adds to its traditions without superseding them — Ludwig V

That's the way it usually works. Math people "create" or "discover", building more or less on preceding results, sometimes obviously but at other times seemingly "new".

It is true, of course, that mathematics often turns out to be useful, but I can't accept that that is its point — Ludwig V

Bingo! :clap:

Top Ten TV series??? — 180 Proof

I'm so old I forget what I've watched. But some are Rake, Lillehammer, Luther, White Lotus, Veronica, True Detective, Fargo, Money Heist, You, Sopranos, and on and on. I enjoyed them all.

I haven't been to a movie theater in at least ten years. With a big screen TV and HBO, Showtime, etc. plus Netflix there is an overabundance of entertainment. I prefer series over films for the opportunity to develop personalities and plot intricacies.

AMC theaters will soon charge more for prime seating. Maybe they'll bring back ushers with small flashlights that theaters had sixty years ago.

No. I meant to say "with depth" — L'éléphant

And you did. My mistake.

I really do not believe that thoughts are even similar to material objects which I also call "things". With talk like this, we create an environment where ambiguity and equivocation are highly probable — Metaphysician Undercover

Yes, and in the quantum world those distinctions could be imperiled. The problem of actualization of potentia brings science and philosophy forcefully together IMO.

Do you really need me to explain to you what I said in english? There are things you could say with depth about the subject besides "Over two millennia have passed with no consensus". — L'éléphant

Maybe in English. And I'm sure you are correct. I suspect most of those things "said in depth" have relevance in philosophical circles rather than in mathematics communities - or anywhere else. It's good to know the limitations of one's reach. I have created and proven perhaps two hundred theorems - but they are virtually worthless, lost in millions more. All said in depth. :cool:

The study of mathematics is not the same as the study of philosophy — L'éléphant

Consensus is vital to mathematics, but from what you say a hindrance to philosophy. When one argues about the reality of numbers, that is not an argument in the realm of mathematical practice. It may have great meaning for philosophers but is seen as incidental to the subject by most math professionals. On the other hand, a philosopher might have difficulty explaining philosophical implications of a theorem picked at random.

There's no debate, Wayfarer. Mathematical objects are quite real for me, though not like Lake Michigan is real. But I do think it's possible that the mysteries of where math "converts" to physical reality (well, I'm grasping at straws here) in quantum theory may shed light on the subject far beyond what philosophers and mathematicians have thought to this time. That's where I see "the profound issue". :smile:

Over two millennia have passed with no consensus. — jgill

Jesus. No disrespect, but if this is all you could say about philosophy, then you don't fit in philosophy — L'éléphant

So, you are saying there has been consensus about the reality of numbers and whether math is created or discovered? I'm not addressing other aspects of Platonic philosophy.

That is what I would describe as a jaundiced view — Wayfarer

I agree. It is. On rare occasions in my career when the nature of the reality of numbers and math came up amongst a group of my colleagues invariably eyes would roll and the topic would disintegrate shortly thereafter. Had I been among foundationalists reactions might have been different.

Look to the future, not the past, pave the roads for all to achieve what they are capable of. I never complained about the idea of affirmative action, only in its interpretations and practice at times.