If there is such a thing, then it is part of "our world". And so the mathematical axioms must be "true to it", in order to be correct. — Metaphysician Undercover

That's just the point. Perhaps we don't really understand "our world" that well. Odd looking axioms should not be cavalierly discarded simply because they are "not true" to our limited view of reality. You misuse the word "correct" IMO.

Instead of writing virtual tomes about the drivel on this thread you should apply your critical thinking skills to actual controversial items like the Axiom of Choice.

So it's completely acceptable to criticize the principles of mathematics when they are not "true of the world", because mathematics is used for purposes which require them to be true of the world. But the game of chess is not used in this way — Metaphysician Undercover

However, modern, abstract mathematics may be more like the game of chess and less likely to describe our world. For a number of years it's been fashionable to move away from the kinds of mathematics that we normally associate with physical reality and into "higher" levels that are increasingly abstract and generalize concepts and processes to the extent that ideas and technicalities peculiar to classical math don't even appear on the radar.

But then I wonder, perhaps what seem like total abstractions really do point to some underlying aspects of reality that we have not reached the point of comprehending. One might think this possibly true with regard to QM, although most of the math used there is fairly traditional. Even oddities like virtual particles are really just mathematical terms in the series solution of difficult integrals. Or so I am told.

Maybe there is a mathematical universe, and somewhere, through all the "chess game rules" mathematicians study, a path to understanding it can be found.

You are certainly reaching far back in history - Ship of Theseus - for your argument.

All the philosophical banter about self has little if any impact and probably pales beside the dramatic shift of perspective Zen provides.

That there is such a thing as a correct opinion, in a sense beyond mere subjective agreement. (A position I call "objectivism", and its negation "nihilism".) — Pfhorrest

Sorry to intrude. I am not a philosopher, but I am not sure what you mean, here. For example, in the philosophy of morals or ethics, what is the "correct" opinion regarding Sophie's Choice?

What I believe is that it's about time for a good dose of healthy skepticism to be directed at mathematical axioms. — Metaphysician Undercover

It must be brutal that few in the mathematical community seem concerned. But I do agree that the axiom of choice is an unhealthy pathology. :cool:

I'm a retired mathematician. There seems to be a considerable overlap with analytic philosophy in the foundations of mathematics and set theory, very abstract topics that occasionally generate what are called pathological examples. However, these areas of mathematics are not extraordinarily popular in my profession. More traditional paths of mathematical thought align with the physical sciences better. Other than that, my opinion is there is very little common ground in the two disciplines.

Thought my post would inspire comments. :cool:

I repeated verbatim what a friend who has practiced Zen for over twenty years said in another forum. I would argue with him that "empty awareness" wasn't possible, for in that state he would have been aware of that state.

Ironic, you deny the individual while presupposing it. Who’s this “one” who learns or has experiences, if not the individual person or subject? — aRealidealist

In fact, I'm denying the sense of self, not the one who experiences.

I would place more faith in neuroscience than Zen or philosophy. :smile:

"Material" here is in the contemporary sense that if it is affected by and/or affects material things — Kenosha Kid

Self-referential definition? Try, ". . . and/or affects physical objects"

↪TheMadFool — Michael

"What I'm particularly concerned about is the ratio between consecutive elements in the set N"

n/(n+1) = 1/(1+1/n) -> 1/(1+0) = 1

Rest easy, mate. Time for a toddy.

This is more a subject for Zen meditation. There one learns, or experiences one's "I" as a fabrication. Instead of "I am aware" there is only awareness. An instant of realization is worth more than a lifetime of philosophical dialogue. :cool:

I wonder how efforts to attract a professional philosopher to this forum are progressing? :roll:

Philosophy is about figuring out how to do sciences. It uses the tools of mathematics and the arts, logic and rhetoric, to do the job of creating the tools of the physical and ethical sciences — Pfhorrest

I think it is unnecessary to even use the word "philosophy" in this regard. It appears you are trying to find a niche, a recognition for something that pervades creative efforts and can be more commonly thought of as combinations of imagination and technical knowledge. You seem to want philosophy on some sort of pedestal rather than admitting it is so ubiquitous, so ethereal that it touches everything but adheres to nothing . . . one needn't even use the term, particularly in scientific research.

I admit, however, that philosophy of mathematics (a truly dreadful discipline IMO) displays the characteristics of both subjects as realized in academic departments.

Philosophy done well guides and facilitates sciences, protects them from the interference of philosophy done poorly, and then gets out of the way to let the sciences take over from there. — Pfhorrest

True enough, if what you mean by "philosophy" is simply critical thinking exhibited by those very familiar with the science and with administrative skills - and not "philosophers" implying some sort of professional status in that discipline. That said, I think you are casting too big a net, calling anyone who contemplates anything a philosopher. At which point the label is meaningless.

When I reflect on my intellectual journey it was specific disciplines that sharpened my abilities to reason, not philosophy in the academic sense. My one senior-level course in that subject did little if anything in that regard.

Pfhorrest's diagram has philosophy at the centre, which is exactly wrong. — Banno

:up: Amen to that!

Philosophy is on the edges because higher learning, thought, and reasoning is produced by it . . . — Outlander

I seriously doubt that. Did Richard Feynman take a philosophy course to learn critical thinking skills?

Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity — EnPassant

Please, please, don't start calling this trash "theorems". And stay away from LaTeX, it's not like playing with a shovel and pail in a sandpile. What you don't know can hurt you. I beg you, leave this disaster zone and return to the relatively safe comforts of philosophical musing. :groan:

One is a quantity, two is a quantity, three is a quantity and so is four — Metaphysician Undercover

Five is a quantity, six is a quantity, and Heavens to Betsy, I do believe there are more! :lol:

You can't jump to infinity and expect the rules of finite arithmetic to apply — EnPassant

Definitely we can, and we always do. Actually, this is all higher math (in contrast to elementary math) is all about. — Andrey Stolyarov

Not "all" higher math is all about. Lots of topology topics, for example, don't revolve about infinities.

There's beautiful piece of math named "Functional analysis" (https://en.wikipedia.org/wiki/Functional_analysis), which works with spaces that have infinite number of dimensions — Andrey Stolyarov

True, but frequently it concerns spaces of functions that are not so elaborate, like complex valued functions of the form

$S=\{{{f}_{s}}(t)={{s}^{2}}{{t}^{2}}+itSin(s+t):\text{ }s\in [0,1],\text{ }t:0\to 1\}$

, where the space extends over values of s (uncountably infinite, but one dimensional). One then considers a sigma-algebra of subsets of these functions and defines a kind of measure of these sets. Then, frequently, a functional, which takes a function to a real or complex number. For instance,

$\mathbb{F}[{{f}_{s}}]=Su{{p}_{t}}\left| {{f}_{s}}(t) \right|$

And so on. Just a simple example. Not everything in abstract math is infinite dimensional, though infinity comes into play.

Whether or not .999...qualifies as a numeral is a matter of interpretation. What I meant, as you seem to have difficulty in understanding, is that it does not signify a number — Metaphysician Undercover

.999... is obviously not a number. It is a numeral. — tim wood

:rofl:

Just a comment about posting math material, symbols, equations, etc. I doubt if anyone here uses it, but MathType is very easy to use and is WYSIWYG rather than coding for each symbol. Under cut and paste preferences, choosing the Wikipedia option and pasting on this forum only requires changing <...> to [...], etc.

Much ado about very little. :roll:

Imagine that there exists a fortune teller, who is able to see what will happen in the future with an almost perfect accuracy. — Mizumono

It's a fun thought experiment, like the Grandfather paradox. My solution? Consider alternate universes created each instant - different paths into the future.

We educate ourselves in a certain field, reach a point where we feel we have sufficient knowledge, then we restart again in a different field — Josh Lee

Been there, done that. So I have a little sympathy for your concerns. I will admit that the second field for me (mathematics) became tiresome at times, frustrating having to do considerable course work and solve problems that had been solved by generations of students, then being tested, etc. Some of those classes took a real and sustained effort to work through, especially if the subjects were uninteresting for me.

But once they were done and I could begin research, academic life changed dramatically and I became free to roam and discover in an exciting new existence. I take it that is how you would like for the entire process to be. I wish that were so, but most of us must work for a living, and society requires credentialing. And playfully creating in an intellectual arena requires laborious preparation. And all those studies in boring topics? Well, you learn the scope of your subject and how seemingly disparate concepts interrelate.

Just babbling - pay no attention to that man behind the curtain. :smile:

I'm not sure that one can conclude that time is composed of individual Planck lengths strung together. My understanding is that measurements cannot be made at these lengths, so we don't know what goes on down there. Maybe continuity over a non-measurable interval, or discrete jumps. Who knows? :chin:

It is TRUELY substantive, if you believe time exists — Pop

In what way is it substantive? Are there physical processes that go one way if time is discrete and another way if time is continuous? :chin:

However, the education system constantly reinforces the wrong mindset where they test and grade us. — Josh Lee

From a student of the mathematician Terence Tao (UCLA):

"245C (1st year grad real analysis) was a course on special topics and was more loosely structured. Your grade was 100% based on attendance. His rationale was, "If you've reached this point, you're here because you want to be here and learn the material. Why burden you." This 'honor system' worked well. I remember spending almost every lunch after class with other students loosely discussing and sketching proofs."

This approach is unusual, but not unheard of. However, Tao's students, who were eager to learn, went through a traditional curriculum to reach this point. Normally, undergraduate classes include some who are not eager, even in their major subject.

Bennigton College has for years made grades optional:

"In addition to feedback during the term, student work at Bennington is evaluated through written reports by faculty members submitted to the Office of the Provost and Dean at the end of each term.
Narrative evaluations summarize an instructor’s view of a student’s progress and are the basis upon which the Office of the Provost and Dean judges the student’s capacity to continue at the College (in addition to evaluating the Plan process and FWT). For this reason, and because the end-of-term evaluative reports make up the largest part of a student’s transcript, students should make a point of examining them closely and discussing them with their faculty advisor. The evaluations reflect the significant aspects of a student’s work. While evaluations are not routinely mailed to parents or guardians, the College may do so at the written request of a student.
The final evaluation appraises the student’s work for the term and is part of the student’s permanent transcript. The final evaluation includes a judgment of Pass (comparable to A+ through C), Marginal Pass (comparable to a C- – D-), and Fail. No credits are given for classes where a student earns a Failing grade. Faculty provide ongoing feedback on student progress throughout the term. They may document concerns about a student’s standing in a course with an Academic Concern Form, which is part of a student’s internal record, but does not appear as part of the student’s transcript. The concern form is part of an ongoing conversation between the student, faculty member, and the student’s faculty advisor."

. . . but the kinds of things you need are the same, differing in quality, not kind. — Pfhorrest

Curiosity and a modicum of intelligence. We all "philosophy" when we contemplate virtually anything. It's a bell curve with mere babbling on the far left and academic scholarship on the far right. But to only contemplate falls a bit short. There seems to be an impulse to express one's self, as well. Is there philosophy without telling others?

Having been a child in the Deep South many, many years ago, I grew up in a white culture which was, to many in the middle class, tolerant of people of color and kind as well, but nevertheless somewhat superior. The lower economic class of whites could be strikingly different. But I observed in my own extended family how prejudices could give way to understanding, even friendships.

I had a much older cousin who lived with his wife and children in an unpainted house in the woods of Alabama, complete with a hand pump in the kitchen sink and chickens and coon dogs in the dirt yard. He was a Klan member, and barely scratched by doing manual labor jobs. After a brief visit as a child I had no contact with him for years.

Upon leaving the Air Force in 1962, my wife and I paid him a visit. He no longer lived in a shanty, but in a nice brick home in a residential area. When he opened the door we saw a middle aged man, neatly dressed. He invited us in, and introduced his best friend, a black man who was a colleague at the nearby BF Goodrich plant which had opened in the late 1940s.

The transformative power of economic progress and equality should not be underestimated. But of course there's more to it than that.

Have you read any Carlos Castenada? — Metaphysician Undercover

His description of the art of dreaming may take one into a truly astounding alternate reality. One of sharp and colorful imagery and experience, where one becomes pure will. But cannot read a printed form. I consider this a mystical experience devoid of a god or supernatural entity. In my opinion, no philosopher could have provided a meaningful interpretation, but a neuroscientist might have. No drugs were used.

This was not a "typical" mystical journey, where one achieves an understanding that one's "I" is an artifice or that "form is substance and substance is form". From that perspective this adventure would be considered a hindrance to jettison on the way to Enlightenment.

Is a concern about whether time is discrete or continuous a truly substantive issue, or is it akin to other debatable subjects that have little relevance to the physical world, like the existence or non-existence of irrational numbers?

I've described my opinion of breeze shooters - and nothing wrong with that, it's a kind of philosophy I suppose, and I should have made that distinction - but what I'm thinking of now is a more academic philosophy focused on a scientific subject like physics.

How much technical knowledge of the subject would enable a serious philosopher to formulate ideas that would be sophisticated and impactful? Could a purely qualitative understanding of the subject suffice, like reading popular accounts of discoveries? In some instances, my answer would be yes. In others, no.

For example, in another forum I watched an intelligent and productive debate between a professional philosopher and a professional physicist concerning Bell's Theorem. Logical inference was the particular concern. On the other hand, I believe some degree of competence in, say, quantum theory, would be necessary for a philosopher to attempt to untangle the mysteries of that subject, when physicists, themselves, have had only limited success. But maybe not. Wouldn't be the first time I've been wrong!

I was saying that metaphysicians apply logic to the mystical revelations. — Metaphysician Undercover

Were I to have an ineffable mystical revelation, how would a metaphysician go about applying logic? A psychologist or neuroscientist might be better equipped to do the job. Or a fellow mystic.

Metaphysicians really apply mystical principles to logic — Metaphysician Undercover

You might flesh this out a bit with examples.

Time may seem discrete because we are unable to measure time duration below the Planck scale.

The act of thinking defines thinking.

Is philosophy therefore more like sitting on the couch and watching tv than it is like brain surgery? — fishfry

Interesting question. I would say philosophy is like sitting on the couch watching a Science Channel show on brain surgery, then chatting knowingly about it with others in the room. Hmmm . . . :chin:

The philosophy of mysticism can be likened to music theory. — 3017amen

I think music theory can be of value in creating music. Can the same be said of the philosophy of mysticism? :chin:

Perhaps this thread demonstrates not that one doesn't care about mysticism, but rather there is little interest in the philosophy of mysticism. I.e., one may care about mysticism, but have low tolerance for the philosophy. Except for a few posters. Just a thought.

the rate of which at any local point . . . — Benj96

The notion of "rate" requires "time". Circular argument. Sorry.

I'm hoping for the sake of the country that Biden still prevails, but it should become clearer and clearer what a stupid decision that was. — Xtrix

I think Val Deming can provide necessary guidance to Sleepy Joe. :cool:

As a mathematician, I often think of time as a variable that generates contours in the complex plane. If one views points in that plane as "events" then the contours represent "histories". No philosophical conundrums there. :cool:

Another great post. Thanks.