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.

It's good someone has re-opened this ancient thread. Carbon's commentaries are particularly interesting, as I received my degree fifty years ago and retired as professor of mathematics twenty years ago. If one contemplates going the distance, only commit yourself if you love to investigate and explore in the discipline. It's not uncommon for a grad student to do well in the coursework, but stumble in the final stretch - original research. I'm sure this is true in philosophy as well as math.

but the disagreement between you and I appears to be as to what constitutes a "mathematical object", or "Platonic ideal" — Metaphysician Undercover

You and I don't really disagree, MU. I enjoy reading your posts, learning of philosophical perspectives I never considered the years I was a practicing mathematician. And I enjoy reading posts by fdrake and fishfry when they pertain to elements of mathematical thought of which I have only superficial knowledge.

If I had known colleagues who were concerned about Platonic ideals, irrational numbers, or transfinite set theory I might have more to offer, but those issues were at best peripheral to our interests. As one of approximately 36,133 descendants of Weierstrass I have been happy using the notions he and Cauchy championed. :cool:

Here's a link to a mathematician's concept of Absolute Infinity. — Gnomon

Thanks. I should have looked it up. For those enraptured with set theory it must seem appropriate that it borders on the divine! :cool:

You know a-priori that a sufficiently computed space filling curve in the unit square is indistinguishable from filling in the unit square in MS paint — fdrake

Then there is the sequence of continuously differentiable functions that converge uniformly on [0,1] to that line segment, but whose arc lengths go to infinity:

${{z}_{n}}(t)=t+i\frac{1}{2\sqrt{n}}Sin\left( n\pi t \right),\text{ }t:0\to 1$

No matter what the magnification, once chosen, n sufficiently large produces a straight line one unit long on the computer screen.

You've forgotten about summation. It might be the case that "4" represents 4-ness, but "2+2" represents a particular instance of the general rule of summation, not 4-ness. — Metaphysician Undercover

You are looking behind the symbols to the mathematics they represent. They are certainly equal in this regard. But if you look superficially at the compound symbol "2+2" and the singular symbol "4" as ink squiggles on paper, they clearly are not the same. But, of course, that's not your perspective. I am simply giving an instance when two things are equal in one sense, but not the same in another sense. You and fishfry can argue ad infinitum it seems. :roll:

If you have some mathematical framework in which 2 + 2 and 4 do not represent the exact same abstract mathematical object, I would appreciate your filling in the details. — fishfry

I'm speaking of the two symbols. 2+2 and 4 are not the same symbols. And I don't appreciate your snide remark. Of course they represent the same mathematical object. I'm not that far gone! They are "equal" in the sense they represent the math object, but are not the same symbols. :angry:

How can they represent the same Platonic ideal when "+" represents an ideal in itself, which is part of "2+2", but not part of "4"? — Metaphysician Undercover

Irrelevant. 4-ness is the ideal in discussion. :roll:

I can concede that "2+2" and "4" are equal but not the same. They do, however, represent the same Platonic ideal. Not being a philosopher, this is as deep as I dare go into the subject. And being a (non-foundational) mathematician this is as deep as I need go. :cool:

Things that may be true, but may not be provable:
1. There is a God
2. String theory

Absolute Infinity does indeed imply that all things are possible, and all possible things are actual. — Gnomon

1. What is "absolute infinity"?
2. Give a rational explanation of how that implies anything. :chin:

Being the same implies being equal, but being equal does not imply being the same. — Metaphysician Undercover

Depends on the contexts of usage. My friend and I are equal (in the eyes of the law), but we are not the same (in the eyes of the law).

This is distressing. I'm beginning to agree with MU . . . :worry:

Obviously the surface area can't be infinite while what it contains is finite. That's just stupid. — Gregory

A simpler example in a lower dimension is $\int\limits_{1}^{\infty }{\frac{1}{{{x}^{2}}}}dx=1$

This is the finite area between an infinite curve in the plane and the x-axis.

This could be reduced to a simpler scenario: Might there be, anywhere in the universe outside our solar system, a Franklin Ace 100 personal computer (I used to have one) not made on Earth? :chin:

Mathematics is ultimately guided by utility, and even those who might seem to be engaged in pure math are doing what they are doing (choosing whichever problems they choose to be working on instead of working on other problems) for a reason, so utility cannot be removed from mathematics. — Metaphysician Undercover

This is true only if "utility" includes fascination with exploring a subject, finding what's behind the next intellectual door, where an investigation might lead, etc. That's been my motivation for many years.

What does Banach-Tarki's paradox do to Euclid's system and proposition 7? — Gregory

The B-T paradox, as has been said several times in this forum, depends upon the Axiom of Choice. Not one of Euclid's. Sometimes the Axiom of Choice is involved in what are called "pathological" examples in mathematics.

I haven't read that paper on nonstandard analysis, but perhaps the "necessary" part refers to the following (Wiki): "The real contributions of nonstandard analysis lie however in the concepts and theorems that utilize the new extended language of nonstandard set theory." And: "Nonstandard set theory is an attempt to generalise nonstandard analysis to cover the whole of classical mathematics."
Too much for my tired old eyes and drifting mind.

Non standard analysis now says that it has rigorously examined infinitesimals and find them necessary, useful, and logical — Gregory

Not "necessary". Useful as an alternative to normal calculus. Logical - in a sense, I suppose.

If you have all the fractions from 1/2 to the infinitesimals (what are they?), you can add them up to make 1 — Gregory

Not sure what you are saying. 1/1 + 1/2 + 1/3 + ... is the Harmonic series that is unbounded (adds to infinity, not 1). The addition of all fractions of the form p/q likewise.

Imagine taking a spaceship (one that forever shrinks) into a banana — Gregory

A+ for original thinking, Greg! :cool:

I don't doubt that there's math in music, but I think the process by which math becomes musically relevant is fascinating and potentially understudied (or I just haven't found the right resources). — Halley

It's somewhat beside the point, but many mathematicians seem to have musical talents as well.

"The question of whether situations and worlds can peaceably co-exist in the foundations of metaphysics is complicated by the fact that world theorists disagree about what worlds are."

Well, that's off to a good start . . .

I read Steven Weinberg's Dreams of a Final Theory some years back and disagree with the premise — fishfry

But he also says this (NYT): ''half-baked philosophy has sometimes gotten in the way of doing science."

:smile:

Here is one direction modern metaphysics is moving. Even as an old mathematician, my eyes glaze over as I try to read it. Those of you more conversant with logic notation might explain it in less technical language. :chin:

We even have an excellent case study, Evergreen University. The academics produced ideologues who produced outrage that produced a culture that produced a cult. — DingoJones

Evergreen State College was designed as an experimental institution, opening in 1967. At one time I knew the Dean of Faculty, Willi Unsoeld, and Pete Sinclair, a professor of English, both having been climbers in the Tetons when I was active there. Willi developed an outstanding outdoor program, but was unfortunately killed in a mountaineering accident while leading a student group. So, initially, the sort of conflict described here did not exist. Evergreen was simply a very unusual progressive college that appealed to certain kinds of students, some attracted to the outdoor program, others wishing to design their own curricula.

What happened after the 2016 election had probably been building for awhile, however. And its repercussions have given Evergreen a reputation that has negatively affected enrollment - last year below 3,000. Not all the faculty supported the destructive aspect of the anti-racist agenda, and a couple who did not and were outspoken were essentially driven from the campus, fearing for their safety. You can find a brief discussion of this in the Wikipedia article.

Cantor's work arose directly from physical considerations. This point should be better appreciated by those who dismiss transfinite set theory as merely a mathematical abstraction. — fishfry

Touché . . . Good point! :smile:

I was curious about what professional philosophers think of modern day metaphysics. In particular, is it possible to earn a PhD in the area of metaphysics? A cursory search yields a number of questionable metaphysical institutions awarding doctorates, but very few, if any, major universities. I did come across this, however, from Stanford concerning their Metaphysical Research Laboratory ( I would not have thought such a thing could exist!):

"The goal of metaphysics, therefore, is to develop a formal ontology, i.e., a formally precise systematization of these abstract objects. Such a theory will be compatible with the world view of natural science if the abstract objects postulated by the theory are conceived as patterns of the natural world.

In our research lab, we have developed such a theory: the axiomatic theory of abstract objects and relations. In many ways, this theory is like a machine for detecting abstract objects (hence the name ‘research lab’), for among the recursively enumerable theorems, there are statements which assert the existence of the abstract objects mentioned above.

Moreover, the properties of these abstracta can be formally derived as consequences of the axioms. The theory systematizes ideas of philosophers such as Plato, Leibniz, Frege, Meinong, and Mally. Our results are collated in the document Principia Metaphysica, which is authored by Edward N. Zalta (Ph.D./Philosophy), a Senior Research Scholar at CSLI. An online version of Principia Metaphysica can be found by following the link to The Theory of Abstract Objects (see below). In published work, the theory has been applied to problems in the philosophy of language, intensional logic, the philosophy of mathematics, and the history of philosophy"

I am completely taken by a machine to detect abstract objects! It sounds weirdly like something from a 1920s sci fi movie that detects ectoplasm through an electrical network involving numerous vacuum tubes. Live and learn. :chin:

Math concerns theorems, which are logically provable. A "theory" in math might be something like Category Theory or Set Theory, areas of study which then contain theorems.

a spin-down positron is just a spin-up electron moving backwards in time. (This is true of all antimatter.) — Kenosha Kid

Nice commentary, Kid. In 1954 I wrote a short paper on this for my physics class in high school. At the time I loved reading science fiction. Of course, the technical details were beyond me, but my teacher, an elderly lady we all loved was impressed. :cool:

There's two problems with this: 1. Hungary was not willfully part of the USSR. 2. Hungarians decided to keep the statue. — Kev

There are no problems here. You have drawn a comparison with confederate monuments. I did not.