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.

Are the statues of Confederate generals put up around 1900 important historical artifacts to you? — Echarmion

I grew up in the deep south before the civil rights era, and as I recall neither I nor any of my classmates paid any attention to the statues of confederate soldiers. But those monuments pale when compared with . . .


In the mid 1950s Stone Mountain was a sort of miniature wilderness, owned by the Venable Brothers. The carving had been half done and left to the elements. Rusty girders swayed and creaked in the wind. I actually did some rock climbing on the lower half of the images. Then, in1958, the state took over and later the area became a kind of Disney world with cheap attractions. I returned briefly in the early 1990s and heard the theme from Star Wars blasting from giant speakers below the completed carving.

It appears the carving will remain intact for the immediate future because of a state law and unresolved controversies about its existence. Some local climbers would like it blasted away and the face opened to climbing. Won't happen.

Buddhas of Bamyan: The Taliban were good at cancel culture



(Wikipedia Commons: https://commons.wikimedia.org/wiki/File:Taller_Buddha_of_Bamiyan_before_and_after_destruction.jpg)

Some of that dust gets into my computer at times!

Or, my computer gets into some of that dust at times!

Very confusing . . . :worry:

This giant statue on top Gellért-hegy in Budapest was a Soviet icon representing the close alliance of Hungary and the USSR. After the Soviets abandoned Hungary thirty years ago there were pleas to tear it down, as it represented oppression by a foreign power. However, cooler heads prevailed and instead of removing it, it was reinterpreted as "Goodbye to Russia!". It remains a beautiful tourist attraction.

What does a theoretical physicist even do if he's not doing experiments? To my mind he could only be doing philosophy — Gregory

It would be good to hear from an actual physicist regarding these comments. We can all speculate. :chin:

The USAF even funded one of my minor research projects that had no military applications. — jgill

↪jgill

Could you please provide more information? What was that research about? — Athena

Could you please provide more information? What was that research about? — Athena

Infinite compositions of linear fractional transformations. Pretty much pure mathematics. :cool: — jgill

It looks like that is being discussed in other threads, but how does it apply here? I am very open to explanations. — Athena

It's not part of an argument. You asked a question and I answered it. It simply illustrates the support the military gave (gives) to scientific research having no immediate military application. I don't consider that a bad thing. But, then again, I served in the military so I guess that makes me a part of the dreaded MIC.

A major force behind the racism in the US was Southern Bells who used media and education to assure their elitist position in the south would be culturally protected at the expense of people of color. — Athena

My mother was a "Southern Belle" and I can assure you she was anything but what you describe. In her later years she helped "women of color" as best she could.

Did you grow up in the deep south? I did.

Wrote a post trying to explain some chaos concepts a while ago. Since you're a meteorologist I'd guess you probably already know it and are making a point regarding chaos being a buzzword most of the time, but just in case — fdrake

I was a meteorologist sixty years ago. A math prof 1971 - 2000. My interest stems from pure mathematics and I've written about extending the iterative process to infinite compositions, mostly in the complex plane. Chaotic behavior crops up, but my main interest is in behavior around fixed points and obtaining striking imagery (not fractals). Indifferent fixed points tend to be the most complicated - even the word suggests thumbing its nose at the mathematician! Thanks. :smile:

Could you please provide more information? What was that research about? — Athena

Infinite compositions of linear fractional transformations. Pretty much pure mathematics. :cool:

Many modern scientists embrace pan-psychism — turkeyMan


How many? — jgill


123 or possibly 159. Not sure which but its definitely one of those. lol. — turkeyMan

:smile: Even one in the hard sciences would impress me. There are probably a few out there.

nonlinear meaning that a small change in the input can yield a huge change in the output. — Kenosha Kid

Roughly speaking, but needs elaboration. Not a definition of nonlinear in the strictly mathematical sense. And what is "small"?. For example:

Linear:
$f({{z}_{1}}+{{z}_{2}})=f({{z}_{1}})+f({{z}_{2}}),\text{ }f(\alpha z)=\alpha f(z)$
$f(z)=Mz,\text{ }M={{10}^{10}},\text{ }\varepsilon ={{10}^{-3}}\Rightarrow f(z+\varepsilon )-f(z)=M\varepsilon ={{10}^{7}}$

Nonlinear:
$f(z)=\frac{1}{z},\text{ }z=10,\text{ }\varepsilon =\frac{1}{10}\text{ }\Rightarrow \text{ }f(z+\varepsilon )-f(z)=.01$

Thinking of it in terms of one observation causing the other to change isn't quite accurate though — Pfhorrest

Yes, that's what I was getting at. Thanks for your comments. :cool: