I have not looked into SIA — Ryan O'Connor

Weird stuff, IMHO. Low priority in the world of mathematics.

. . . that one person should not both write the textbook, and teach from it, and test the students on whether they know their stuff. But, arguing for or against that is going to be a topic of one of those later threads — Pfhorrest

Depends on that person. Is it a good text? Does the prof profit from its sale to his students? Is the prof fair? Not an easy yes or no here. But carry on.

I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. — Ryan O'Connor

What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way. As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. Something like Bergson's notions from a century ago. Here it is.

A separation of research, teaching, and testing branches of education
is therefore analogous to a separation of legislative, executive, and judicial branches of government — Pfhorrest

Questionable analogy. In academia one person may have responsibilities in all three, in fact usually does. Not so in government.

Old age is not for everyone. And you never know what's going to go wrong for you or right for you in life. At age 84, my only advice is keep an open mind and stay physically and mentally active as long as possible. I recently returned to my fitness gym, after getting both vaccine shots, and it was a pleasurable experience. But then I have been athletic since the age of 17 and have kept it up. Don't boo hoo when you begin to decline and can't kick the soccer ball like you used to. If you keep at it you may be pleasantly surprised in old age, for your perspective will change. I too have scoliosis and severe arthritis. But four ibuprofen tablets and an acetaminophen taken an hour before exercise can work wonders.

If you make it there, that is. It's a crap shoot. :cool:

You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. — Ryan O'Connor

I know what you mean here, but reading it makes me uneasy.

LoL. I'm sure the philosophy students in the room were aghast. — Ryan O'Connor

That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. :smile:

You don't know anything about it.

Yet you have persistent critiques of it.

How do you do it? — GrandMinnow

With a certain aplomb. I admire his spirit while avoiding his critiques. :cool:

To me it is concerning that the foundations are so disconnected from the applications — Ryan O'Connor

It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely.

I've mentioned my own experience before: as beginning grad students sixty years ago we were required to take an introductory course in set theory (foundations). Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." I wasn't and I didn't. But I ultimately went into classical complex analysis. Had I gone into a more abstract realm of math I might have needed it.

Could this be an indication that further foundational work is required? — Ryan O'Connor

It depends on which mathematician you ask. Let's hope not. It makes little sense to ask anyone outside the profession.

I'm no expert, but my impression was the math moved toward being totally mechanized, totally formal, totally computer-checkable — norm

And I'm sure Norm would agree, that movement would drive most mathematicians out of the profession. It can't be emphasized enough how much mathematics depends on intuition, imagination, inventiveness, and a spirit of exploration. Devising and proving theorems is an art form.

How can ∞ represent a point? — Metaphysician Undercover

The north pole of the Riemann sphere. But carry on.

Unfortunately, there are math teachers who criticize a student who solves a problem in a novel way.

Before I retired twenty years ago there were discussions in the math department about race and subject matter. We agreed that whenever a topic came up in which a minority had made a significant contribution we would mention that fact. But twisting the subject matter around into some sort of ethnomathematics was not considered.

Hi ! I'll private-message you about that. — norm

Oh boy they're gonna gossip about the rest of us! — fishfry

Not so, my friend. Norm is mathematically authentic, as are you and fdrake, and I will probably learn something from his posts, as I have from the two of you.

I'm of two minds about revealing anything about the expertise of math people on this forum. I realize the knowledge may intimidate some others and dissuade them from contributing their ideas. Or it might have the opposite effect of encouraging attacks on academia. Oh well, not a big deal.

After years of formal study (proof writing), I still would argue that intuition is primary and that math is a language — norm

What's your background, Norm? Another math proof on the forum? If so, welcome. :smile:

Connected with the ideas of "complex" and "complicated" in proofs is the word "elementary", which, when used by a mathematician, usually does not signify "simple" or "easy", but refers to the level of a mathematical argument, specifically that that argument involves only basic concepts in the specific subject area. Some elementary proofs are very complicated. Others, not so much.

"Non-elementary" usually means the argument involves more advanced concepts and results, and may actually be fairly straightforward and uncomplicated - or not. :cool:

In your professional research, did you have the feeling that you were investigating aspects of truths outside yourself that you were trying to find out about? — fishfry

I'm more or less continuously involved in light, unimportant research these days. As an example, I've defined a variation of linear fractional transformations in which the normally constant fixed points are themselves functions of the underlying variable. Then I search for criteria that produce convergence upon iteration. I definitely look for "truths" outside myself, and it's this exploration that's fascinating. Knowing that such truths depend ultimately on axiomatic structures has no bearing on my minor investigations, and the very thought of moving around symbols in a formal game is anathema.

I was once a rock climber and it was the delight of exploration that was intensely compelling. :cool:

One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. — fishfry

It's true, math is a social activity, but I bet a lot of it exists without a preponderance of mathematicians even being aware of it, much less agreeing it exists. A single practitioner can have an original idea, one that assumes mathematical existence at that moment.

Wiki: "Existence is the ability of an entity to interact with physical or mental reality."

On the other hand, it does take that kind of recognition to establish the importance of a mathematical idea in the mathematical community.

Once again, a topic of limited interest to practitioners. :smile:

I'm lost on why the problem is such a big deal — Darkneos

It's not. Seems like common sense.

Is this thread intended to find something akin to the p vs np problem (related to complexity in computing)? — emancipate

Good question. I think there is confusion regarding complex in a technical sense and complicated in a more general sense. I know of no "scale" describing levels of complication in a mathematical proof, although when one conjectures a proof path it may be apparent that low or high levels of complication might ensue. And maybe there are practitioners who speculate "Why this looks to be about an eight on the scale of complexity!"

Again, a discussion of mathematics that mathematicians pay little attention to. But fun for amateur philosophers. :smile:

I have no use for category theory, but it does attempt to generalize areas of math that have similarities. I fear your knowledge of mathematics is so minimal that we are not getting anywhere here. Since I have been a mathematician (over fifty years) the subject has grown so dramatically and is so complex now I understand little of it myself.

A lot of people are baffled by this topic for some reason, where I see this in a sense very apparent in mathematics nowadays to talk about the sort of lack of categorization, that can even at all begin or take place! — Shawn

Category Theory

(and perhaps you would even agree that the greatest value which y never reaches is 0) — Ryan O'Connor

There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches. Your attempts at the philosophy of mathematics may never bear fruit if you consider this a cogent statement.

Try this one: https://www.reddit.com/r/PhilosophyofMath/

Of course, Stack Exchange in mathematics is sort of a forum, and the participants include very knowledgeable professionals, although not so many mathematical philosophers. Very few on TPF have any graduate math experience. Exceptions include fishfry and fdrake, both of whom are up to speed on modern math topics including foundations, which overlaps philosophy. I retired over twenty years ago and am not as reliable. :sad:

Sure, but this is quite a conundrum towards the notion that everything in mathematics should or is determinate. — Shawn

The question of whether mathematics is created or discovered has been around for a very long time. This, perhaps, is not precisely what you are asking, but close. An act of imagination may lead to interesting results that are determinate, but the original thought might never arise. Then all that follows will not exist. An original thought is required to set the train in motion. Is that thought determinate?

Let's suppose a math guy has that original thought, and a process unfolds in a determinate manner, then someone else has an original idea that influences the direction of that process. The second guy is part of the process, but his thought may not be determinate.

Most mathematicians pay little to no attention to issues like this. I never did. :cool:

Is this the result of one of your mathematical formulas ? — Pop

Yes. I've developed the elementary theory of infinite compositions of complex functions in the complex plane - a subject of very, very low importance in mathematics. It's a kind of dynamic systems. The image arises from a BASIC program I wrote some time ago.

David Chalmers, a philosopher and social scientist, wrote a nice article on weak and strong emergence a few years ago.

Scientists don't need to think about philosophy all the time but they do this often while doing theoretical physics — Gregory

No argument from me.

Penrose's conformal cyclic cosmology says, according to his interviews, that the universe will expand until "it no longer knows what size it is". You have to think philosophically to unpack what that means — Gregory

And that is precisely what the Nobel Prize mathematician/physicist, Penrose, tosses at us. His is philosophical speculation by a revered scientist - not a non-scientist philosopher. That was my point. To philosophize in modern science one needs a science background.

The point being that philosophers can help scientists with conceptual orientation? — Banno

Yes, those poor scientists need a course in critical thinking skills taught by a philosopher. :roll:

Their ideas bring paradigm shifts which allow scientist to frame theoretical matters in new ways. Philosophers don't do the measurements but measurements can never stand alone without conceptualization of them and those concepts have much to do with what is discussed in philosophy — Gregory

You may have a point, but I am unconvinced. You would have to demonstrate this philosophical prowess in examples in the modern world, not olden times.

Namely, the decreasing amount of philosophers or scientists that exert too much of an effect on a field by making their name known.

Gödel did this to mathematics. — Shawn

He brought to light a fallibility in the subject, but it's not a concern in much of mathematics. However, in the future it may turn out to be very important. Who knows?

Weak emergence, weak relationships > The emergent properties apparently aren't very strongly connected to the parts — Kaiser Basileus

"Weak emergence" is a technical term and does not mean "weakly" emergent. It implies the equivalent of a computer program producing an unexpected and largely unpredictable outcome. Thus the image I posted.

. . . but science can't do without philosophy whether you like it or dislike it — Gregory

Progress in science requires lots of speculation by scientists, and some of this could be called philosophy. But to stipulate that philosophers untrained in science can trigger scientific revolutions is a stretch. :roll:

Emergence is identical to relationship — Kaiser Basileus

Here is one of my examples of weak emergence. What is the "relationship"?

Human thought and ingenuity is paramount in creative mathematics. There is no way to determine this in advance. Lots of "aha!" moments. :cool:

So what are you on about jgill? And I mean that question literally; I have no idea what you're actually objecting to. Incidentally, no, calculus doesn't give us the GH paradox... broken intuitions do. I also find it a bit strange to claim that calculus is used to define the object; rather, it's used to analyze the object (surface area/volume in this case — InPitzotl

Math 631 (Algebraic Geometry) (U of Mich):

"Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should either already know or be concurrently taking commutative algebra (Math 614). Students should also know the basic definitions of topology — we won't be using any deep theorems, but we will use topological language all the time. Basic familiarity with smooth manifolds will be very helpful, as much as what we do is the hard version of things that are done more easily in a first course on manifolds. Undergraduate students intending to take this course should speak to me about your background during the first week of classes."

This description speaks for itself. Correct me if I am mistaken, but it appears you have tossed in AG to impress the readers of this thread. If you are indeed a mathematics professor and feel AG is necessary, then I would understand. Are you? I was one for many years and we never had an undergraduate course in AG, although some schools do. GH always came up in a standard calculus course. Tell me where you are coming from and why you found it essential to define GH this way.

By the way, you should now go to the Wikipedia article on GH and inject your considered opinion. It's a nice piece and never mentions AG. You apparently think it should. Again, if you are or were a professional math person and have strong feelings about this I will understand.

From the link above: "So, I don't think anyone has addressed the question posed in the title; but, is complexity in mathematics in your opinion determinate?"

No. No more so than complexity in human thought is determinable.

What do you mean this has nothing to do with algebraic geometry? — InPitzotl

Whereas one can describe the collection of points in 3-space comprising GH with the zeros of
$P\left( x,y,z \right)={{z}^{2}}{{x}^{2}}+{{y}^{2}}{{x}^{2}}-1$, the paradox of GH does not emanate from that perspective, but from elementary calculus. Why even bring varieties up since it is irrelevant to the issue being discussed, and participants of the thread might well be familiar with the rudiments of calculus, but have little acquaintance with algebraic geometry?

A penny for your thoughts — TheMadFool

I'm glad to see you've left mathematics behind for the moment. :cool:

Gabriel's horn is an object defined using algebraic geometry. Algebraic geometry defines points in a space using coordinates using number lines. Number lines are defined with real numbers. — InPitzotl

Nonsense. This has nothing to do with algebraic geometry. G's Horn is elementary calculus. :roll:

You guys should just let this go and get back to epistemological metaphysics where accuracy is optional.

V = pi * (r approaching zero) * (r approaching zero) * (h approaching infinity) , (r approaching zero) * (h approaching infinity) = 1 — TheMadFool

Suppose r=1/n and h=n^2. Then V -> pi. You are not describing Gabriel's Horn.

If infinity = z then, — TheMadFool

This is mysterious. One should make pronouncements about topics familiar to one.

(Not being a philosopher, this makes me wonder if some of the "sophisticated" philosophical arguments on the forum are any better) :roll:

Possibilities in theoretical physics should be left to theoretical physicists, some of whom can be considered philosophers.