Intuitively, Smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved.

Until we take notice of the reality of how space and time are actually quantized in real discrete units, these attempts, such as limits and infinitesimals, will remain ideals of theory which do not adequately represent the quanta of reality. — Metaphysician Undercover

Perhaps space and time are not "actually quantized in real discrete units".

To find meaning in what one does or is, regardless of the degree of volition giving rise to the doing or being. As an example, you might be in the Army and be assigned to a hospital ward to assist with the disabled. At first you find this task upsetting, possibly revolting in some instances, but as time passes you have a change of heart and find deep meaning in your work.

Another example, you are young, casting about for a purpose in life, and an older friend gives you an amateur artist's kit while you are recovering from an injury. You find you have a natural talent and love what you've discovered, making it the vital theme of your life from then on.

In both examples meaning arises and is embraced. Once it is triggered, you are hooked.

That is to say, why we start any endeavor or project (or choose to continue with it or end it) is shaped continually by a deliberative act to do so — schopenhauer1

is it moral to cross that boundary intentionally, or to be entertained by real life conflict? — Benj96

Shadenfreude is one aspect.

Truth table

Restrict the philpapers results to metaphysicians in the target group of academic philosophers - 372 respondents - and the number who advocate idealism goes up to almost 7%! The number advocating realism rises to 84%.

Make of this what you will — Banno

I love it when philosophers dabble in statistics. It's fun to visualize a group of academics sitting around a coffee table in the philosophy faculty lounge tossing numbers at each other. One says, "Hey, that's wrong - but in an interesting way". :cool:

I see from my readings here that my thinking needs modulation by your robust brand of optimism. — ucarr

:rofl: .

Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it — Infinity Is a Beautiful Concept – And It’s Ruining Physics - Max Tegmark

Lots of mathematicians don't use the infinity symbol except in a limit sense. The idea of some entity called infinity existing doesn't normally crop up in classical analysis, for example. Tegmark cites the expression $\frac{\infty }{\infty }$, but elementary calculus shows ways around that (in very elementary settings), stating the expression is meaningless. If I were to say $\underset{n\to \infty }{\mathop{\lim }}\,f(n)=\infty$, I only mean that as n increases so does the function, with no upper bound. I don't mean it ultimately ends up at a magical point at infinity.

All this is very simple. I've read that Hilbert spaces used in QM must be infinite in order to preserve continuity and completeness in some circumstances. But there are ways of dealing with these concepts in constructive mathematics. I suspect the noise made by math foundationalists regarding cardinalities and "squeezing in" numbers in R by forcing and such things might make physicists uneasy - if they even care. I don't. :cool:

His last participation was about a year ago. He had made some comment about pursuing a "Pot of Gold" with his guitar. :cool:

Since axioms are produced by mathematicians who practise pure mathematics, and those people who apply mathematics have a choice as to which axioms are used, it would appear like we ought not use axioms like these, which necessitate that aspects of reality will be unintelligible to us. Instead, we ought to look for axioms which would render all of reality as intelligible. — Metaphysician Undercover

Those who apply mathematics normally are not deeply conversant with its axiomatic structure. Most mathematicians - in the pure realm - are not either. It's dreadful stuff, but some treasure it. :cool:

There was a member here, active a couple years ago, I can't remember the name, but a self-proclaimed physicist who was big on this time reversal stuff. — Metaphysician Undercover

@Kenosha Kid. He was a Q-physicist who left the profession to play his guitar, as he explained to me. He liked Transactional physics.

Brian May is an example.

:up:

However,

The hard problem really boils down to "What is it like to be another conscious being?" — Philosophim

this doesn't seem quite correct. One could argue that someone with MPD experiences just that. But, you might say, they shift from one to the other, so being in one state at a time. I would even question that assertion. I have actually had a meditative experience in which I was able to be myself and another simultaneously for a few brief moments. No, I'm not crazy. As a matter of fact whenever we talk to ourselves we indulge very slightly in this experience. But I'm setting up a strawman here, so I'll leave it.

Don't know how large a leap that is — TiredThinker

The truth table for that statement has him taking no chances.

Yes. You did mention him. I found some of his papers on researchgate.net, to which I also belong. This guy is older than me!! How can that be?

I'm not conversant with his areas of expertise and that is usually a substantial impediment in mathematics. But I appreciate you pointing him out. :smile:

This Article is intriguing. At first I thought they had found a way to reverse time in the quantum world, but rather they rejuvenated a photon, taking it back to a previous state.

The mathematics involved is probably linear (much is in the quantum world), since most non-linear systems are not reversible. Who knows? :chin:

as jgill says, many maths educators are not the least interested in the philosophical question. — Wayfarer

I must admit that in my old age one philosophical issue does interest me: where does the set of potentials created by a "new" concept or discovery reside? And what triggers actualization in the arena of mathematical knowledge? My mathematical interests have always been in infinite compositions, and I see a structure therein that might model this process. :cool:

Looks like it could be 1/2. I used to teach stuff like this many years ago, but haven't indulged since. Could be 1/3 I guess. If you draw a diagram of paths possible, only one is favorable, with probability =1/2. @fdrake could step in here with clarifying comments. :cool:

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: