I only wish to tell you that with your background, every single thing that's been discussed on this forum that you think is beyond you, is actually trivially within your capabilities and knowledge. — fishfry

Fair enough. When I speak of a topic being "beyond me" it's a cop-out for not having the mental energy at my age (83) to study it, or just a complete lack of interest. I appreciate your comment.

Every so often, however, something a bit out of my purview will intrigue me and I will make an effort to understand it. For example, a couple of years ago the notion of a functional integral sparked my interest, having read of Feynman's Sum of All Paths concept. My brief exposure to the concept fifty years ago was shallow and uncompelling.

That was a delightful exploration, starting with the basic Wikipedia definition, and I wrote a short math note about functional integrals in spaces of complex contours. I enjoy writing math programs, especially graphics, and I came up with some nice imagery. That was fun.

I should not be making dismissive comments about set theory. You, fdrake, Nagase, and a few others have clearly explained ideas in this subject, and it is a powerful link between math and philosophy, and a vital part of the mathematical galaxy. I apologize, and if I slip up in the future you should nail me!

Most of my research efforts have been in classical analysis, very basic dynamical systems in the complex plane, trying to determine convergence/divergence of certain sequences. At one time this was a popular topic, but modern analysis has moved the focus more toward algebraic systems and generalizations.

But I remain attached to the old-fashioned, nuts and bolts, stuff. For example, my latest efforts concern the iteration of linear fractional transformations (f(z)=(az+b)/(cz+d)) when the attracting fixed points are functions of time and are no longer "fixed". Like predator and prey, do the iterates "catch up" with the roving attractors? Modern theory dealing with LFTs is more geometrical and algebraic.

OK. Enough rambling. Thank you for your comments.

The social, political, and military chaos of the 60s dwarfs anything happening today. We got past that. We'll get past this too. — Hippyhead

58,220 American casualties of the Vietnam War. Three assassinations of US leaders, including POTUS. Rioting, murders of black civil rights participants, murders of demonstrating students. The rise of cults. Then Watergate . . .

I agree. We'll get past this. At the time, in the midst of the Cold War, it seemed America was falling apart at the seams.

2. Behavior of matter in the brain is changed from what it would have been if governed completely by . . . — Francis

Sorry. Nit-picking here. But how can something change from what it would have been? :chin:

Nice art works, Bob. Welcome to the forum. :cool:

Higher education is under considerable pressure from several different directions. Even before the pandemic hit the public attitude was shifting away from the requirement or even advisability of a college degree. Now, with student enrollment dropping, the financial crunch is enormous, and some institutions may fold. Those that remain must, and probably should, become heavily involved with anti-racism efforts, which may lead to internal conflicts. And then, who wants to pay full tuition at an elite school like Harvard only to be stuck in front of a computer screen off campus?

Quite a few recent PhDs find themselves cobbling together a meager living by adjuncting with several institutions. And those who are fortunate enough to gain tenure may find themselves in academic rat races to publish and secure external funding. The environment within an academic discipline can become toxic at times, with pressures to beat a colleague to a new result encouraging ignoble behavior and animosities. But if one really loves doing original research these impediments are manageable. And if you think you have great new ideas, these must be evaluated by experts in an academic environment with one's peers. Not on internet forums.

If you want to make philosophy (or any other academic discipline) a career, you'd better really love it and be prepared to make the sacrifices required!

you often announce your ignorance of virtually everything outside of your specialty, and most of the standard undergrad math curriculum. — fishfry

Really? I've taught analytic geometry, the calculus sequence, advanced calculus, intro to real analysis (grad), point set topology, statistics and probability, history of mathematics, mathematics of finance, metric spaces, intro to computer science, complex variables, differential equations, special topics in analysis, math for engineering technology, linear algebra, etc.

I have taught neither abstract algebra nor set theory. The one faculty colleague who was a set theorist became more of a math educator and journal editor. We never developed a set theory course due to a lack of student interest as well as the fact that most of us were not competent in that subject. As you have seen.

I have not kept up with more than a small handful of the myriad of directions math has taken during the last century. I got my degree fifty years ago and teaching, administrating, and on and off research kept me busy, along with a family and a serious outdoor avocation. One makes choices.

Please stop the ad hominem attacks.

So we need to develop another dimension of time to account for this substantial change which occurs at the present, and relate it to the other dimension of time which is supposed to be a continuity through past and future — Metaphysician Undercover

Let me think about this. I wrote and posted a note on complex time recently that expresses a "real" and an "imaginary" time variable. But not in the way you describe. Interesting.

What do you mean to communicate here? — fishfry

What is upsetting you? I have simply stated something I haven't done. There are no unstated jabs at modern mathematics. And yes, I am curious to see what that might look like. Nagase gave a nice reply to a question I raised. Since I once went through the process of attaining the exponential function through set theory I am curious how one would prove something like the fundamental theorem from "scratch". But I am not curious enough to work through all the details.

Do you wish math would go back to the mid nineteenth century before the age of formalization? — fishfry

Of course not. I know you think I'm a dinosaur of a mathematician, and I admit I am. I have even avoided holomorphic functions at times in order to investigate more general non-linear conditions in complex dynamics. That really puts me out to pasture. But you know what? It's been a delightful journey and it still is.

I apologize if I have offended you. Let's agree that modern set theory as well as other modern areas of mathematics are very important subjects that some old timers may not fully appreciate. :cool:

In other words, it appears to me like mathematicians have posited a type of "object" which is metaphysically unacceptable. — Metaphysician Undercover

From Wikipedia: "The strong, classical view assumes that the 'objects' studied by metaphysics exist independently of any observer, so that the subject is the most fundamental of all sciences. The weak, modern view assumes that the 'objects' studied by metaphysics exist inside the mind of an observer, so the subject becomes a form of introspection and conceptual analysis."

I've never come across, nor contemplated tracing the proof of something like the fundamental theorem of calculus back to its set theoretic roots. As I've mentioned, in a set theory class sixty years ago we did define the exponential function, however, from the PAs.

Our experience of time indicates that there is a substantial difference between future and past, and therefore no necessary continuity of substance at the present. — Metaphysician Undercover

It seems to me your "therefore" does not logically follow. The "substantial difference" requires a temporal distancing. I suppose you discard elementary calculus, with its notion of time continuity, and its many physical applications. Maybe not. Following arguments in philosophy is sometimes like trying to separate the filaments of cotton candy.

Rule number one, you must count distinct objects, you cannot count the same object twice. No matter how many times the same object appears in front of you, you still only have one object. — Metaphysician Undercover

But what is an "object"? And what is a "metaphysical object"? Is there an overlap? :chin:

As an old retired mathematician not conversant with contemporary set theory, could you clarify your recent posts by explaining how they relate to common, everyday theorems arising in classical areas of mathematics, like classical analysis. Are you saying the collection of proofs of all such theorems in this context are computatively enumerable? :chin:

The upshot of all this is that, in my opinion, constructivists should resist the temptation of reducing truth to provability. — Nagase

Not only intuitionists (who would require an actual construction), but other practitioners as well. :cool:

↪jgill
So what was the point you hoped to make? How does it relate to the physics of a Big Bang universe? Break it down for me so that I might understand. Give us an example in a physical context. — apokrisis

No relation. Your description of a "tipping point" in physics caused me to see certain neutral fixed points in complex dynamics from that perspective. Moving a tiny distance away in one direction gives a value that iterates back to the fixed point, whereas moving a tiny distance away in another direction gives a value that quickly iterates far away from the fixed point, although it may eventually return. Of course, a repelling fixed point would send any point close by further away.

I've written over 175 mathematics programs, most focused on graphics illustrating mathematical concepts, but not being a subscriber on this forum means I can't upload the graphic imagery. A simple vector field in this case would show immediately what I have described. But I realize this is not the topic of the thread, so I apologize for deviating :yikes:

Are you a physicist? You seem very knowledgeable.

How does having a higher IQ alter or modify one's behavior? — Shawn

Not sure you are saying what you mean to say, here. :chin:

The Peace Corps, maybe? Or some other overseas volunteer services. Sometimes those experiences can be very gratifying and clarifying.

Yes. But what if this non-linear sensitivity is being regulated by a parameter that is a reciprocal relation such as y=1/x? And so yx = 1? A tiny tip one way is yoked to a tiny tip that compensates. Unit 1 has been fixed as the identity element, the common departure point. The indifference lies in now giving it any particular value to denote some quantified scale. It is now always just a generalised quality - the way an identity element behaves as a symmetry awaiting its breaking. — apokrisis

Sorry, we must be talking past one another again. I have no idea what you are saying. Here is a parabolic LFT having a neutral or indefinite or indifferent fixed point in C. Depending upon the value of K one gets the behavior I described before.

$F(z)=\frac{\left( K+\alpha \right)z-{{\alpha }^{2}}}{z+K-\alpha },\text{ }F(\alpha )=\alpha ,\text{ }F'(\alpha )=1$

Seeing that I have no future, I don't see why I shouldn't just study Philosophy. — thewonder

Sorry, but this sounds like the worst possible reason for enrolling in graduate school. Probably guaranteeing dropping out.

Why not join the military and let them pay all expenses for career education? I got paid to go to school and become a professional meteorologist, and a friend became a lawyer and is now a district attorney, another became a physician.

But I bet you have a low opinion of the military! :smirk:

In the world of pure mathematics, a vector field in the complex plane describing a function F(z) having an "indifferent" fixed point a=F(a) might show the enormous differences of displacement as z=a is "tipped" a tiny bit to one side or the other. On one side it might immediately be iterated back toward a, while on the other side it might be sent far out in the plane. It often happens, however, even in the latter case the iterated point will eventually be brought back to the neighborhood of a. So this would be a kind of symmetry breaking in the complex plane. This happens with what is called a parabolic linear fractional transformation, as an example.

I like this (from Wikipedia) illustrating AC:

"Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice."

The axiom of choice serves to make infinite sets behave as they should and is therefore the more natural choice than its negation — fishfry

Nice defense of the AOC ! :cool: I think that those who spend time studying set theory and foundations have a much deeper appreciation for those subjects than do those of us who are briefly exposed to it and move on to different topics. It may have been a shift towards foundations the math department at the U of Chicago initiated back in the late 1950s that caused a rift with the physics department and resulted in physics students being required to take all their math courses in the school of physics. But I don't know how long that lasted; it may have been a temporary policy.

And without the AOC or something similar we wouldn't have non-measurable sets. :sad:

OK. Thought you meant rolling along its circumference.

A sine wave is the trace carved out by letting a disc roll for a length by a mark on the circumference. — apokrisis

Wrong. It's a cycloid. Far more complicated than a simple sine wave, which it resembles superficially. :roll:

In consciousness, you have metaphysical phenomena. Universal things like the experience of love, the will to survive, self-awareness, wonder, etc — 3017amen

So are sensations and emotions part of the "truth" of the universe? Are they metaphysical, like String theory with its mathematical context? I remain curious about modern set theory. Has that been inspired and gifted to mankind in preparation for a future revelation about a truth of the universe that will indeed embody what seems now to be only faintly related to our physical world? :chin:

I would be curious to see if any of the topics discussed here as metaphysical would arise as "abstract objects" from the algorithm at the Stanford Metaphysical Lab. Anybody here from Stanford?

Metaphysics is generally going to be in advance of some mathematical notation — apokrisis

But not entirely, as seen in applications of non-Euclidean geometry. Perhaps there is a "mathematical universe" behind the physical one we see and a mathematician's subconscious makes a connection and forms a pathway for those ideas to enter our consciousness (along with a lot of garbage, of course!), and this in turn sparks metaphysical flames.

Mathematics can not say what the truth of the universe is — fishfry

And that presumably is what metaphysics is all about. Or is it? :chin:

And the fundamental physicists are today's actual metaphysical community. — apokrisis

That's what I would have guessed.

Stanford has a Metaphysical Lab in which an algorithm generates "abstract objects". I have commented that Leibnitz's mathematical infinitesimals may be considered a kind of metaphysical actuality, but others disagree. There are now rigorous logical structures embodying them.

A major obstacle here is there is no clear definition of metaphysical. Some might say that String theory is metaphysics, others might disagree. Mathematics beyond infinitesimals, or parts of mathematics could be considered metaphysics. I consider transfinite set theory and all its offshoots to be a kind of metaphysics, and there are oodles of proofs of theorems therein.

However, I am handicapped by having been a mathematician and not having been a philosopher, so take my comments with a salty grain. :cool:

Life goes to infinity? Perhaps explain the diagram.

Do you understand the difference between "having the same value", and "being the same"? — Metaphysician Undercover

Two diamonds have the same value, but are not the same. They are equal in value. They appear equal when viewed without high magnification. Just babbling, pay no mind. :roll:

From Zen: Form is emptiness and emptiness is form:chin: .

You joke, but the US military has admitted there are UFO’s — DingoJones

True enough. But they may not be aliens. :cool:

so many begin their discussion with "let's first define our terms". — Banno

Alas, were that so . . . :roll:

Are you and I the same, just because we're equal? — Metaphysician Undercover

Good point, MU. :nerd:

You think Covid-19 is just a virus? Think again, Earthlings! :scream:

Who knows? I wonder if there is any truly empty space that is devoid of fields, like the electromagnetic field. :chin:

and one or two crazy professors go as far as forcing. — Nagase

:smile:

Thanks for your comments!

Sixty years ago I took a course entitled "Introduction the Graduate Mathematics" using Halmos' book on naive set theory plus handouts and lectures on the Peano Axioms and PA. I don't recall any critiques of PA, only using them to construct some elementary mathematics. I think our assignments got us up to the exponential function. We discussed Gödel, but only in passing. I wonder what students are required to study nowadays. Perhaps the math in math programs and the supporting logic in philosophy programs, with not a lot of overlap.

Whatever happens, I definitely do not want to spend more than three years for the PHD. I have to make sure that is possible before I make any decision. — Eros1982

From my experience, this may be possible if you take a full complement of courses and don't get stuck with your research project. The PhD is a research degree. If you teach part time it might take longer. My father received an MA in math in the late 1930s, was a statistician for the fed reserve and War Assets Administration, and returned to grad school at the U of Texas for a PhD in business statistics - which he completed in two years. In lieu of course work he was given credit for experience.