I'm curious. As a retired prof I don't think it's accurate.

Here's another:

Length & Complexity

I'm wondering about reaching out on somewhere like Stack Exchange for more information or how to answer it — Shawn

That's a possibility, but be prepared for some confusion about what you are asking. You're in a different ball game on that forum.

A machine for ascertaining the length of the shortest proof of a theorem doesn't involve anything like oracles — TonesInDeepFreeze

A posteriori I assume? Not a priori.

I find it interesting as a mathematical question onto itself. — TonesInDeepFreeze

True enough. Beyond my pale.

I sort of dread the notion of making mathematics more formal. :scream:

Interesting ideas, but to my mind it's putting the cart before the horse. I don't see how this sort of knowledge would help prove a theorem, which is the essence of mathematics. But then the very idea of theorem proving as a programming exercise is foreign to me. However, I use the computer all the time for imagery and examples which may point the way forward in my deliberations. My age, I suppose.

you can do just that. You can take two separated points on a plane and "glue" them together, making them one and the same point. — SophistiCat

Refresh my memory, how is this done?

Edit: OK. Quotient topology. From Wiki: "Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space." Your reference to "points on a plane" confused me a tad.

Just one of a great many math things I was unaware of! :chin:

Nice :up:

(and unfortunately, few mathematicians are really familiar with any of them). — alan1000

Is this opinion or fact? Please state your references if the latter.

Well, yes. I'm concerned with complexity of a theorems proof or what I think is how you can gauge complexity in mathematics, through determining how simple the proof is for any theorem. — Shawn

I'm thinking again of having proven a theorem, putting into a computer algorithm, then counting symbols to ascertain "complexity." I guess that would be some sort of definition, but why one would wish to do that is not clear to me. I suppose one could say, "Oh, Gill's proof of Theorem X has complexity 32, whereas Kojita's proof has complexity 56." Then what? :roll:

You've brought this up before. Let me get it straight. You are not talking about deriving the proof of a math theorem, you are asking that if I were to conjecture a theorem, then prove it, and then put my proof into some sort of computer algorithm - which is just another way of writing it out - is there a way to determine the "complexity" of my proof by examining the algorithm? Is this what you mean?

which leaves open the possibility that a theorem may be complete, yet possibly incomprehensible. — Shawn

Are you saying a theorem is complete if it has been proven?

I guess I am confused about something. — Daniel

Very few will admit that on this forum. I admire your honesty. I think you mean limits to be bounds. Do you? Like a circle in the plane vs a parabola in the plane.

More, if I had it to give — TonesInDeepFreeze

Blessings on thee, kind citizen. :blush:

Would this thread be considered good physics or bad physics? I can't tell because its a foreign language for me. :chin:

Taxpayer here. Make math, not war — fishfry

Math maker here. Where the hell is my tax refund!! :confused:

Since I last checked 137 math papers have been uploaded to ArXiv. The rate of research in math is staggering, as are the number of specialized topics. It looks like algebra and its offshoots account for about 26% of the total. Only 2% in complex variable theory, my general area, outdone by logic/set theory/p-s topology/formal math - one category - which comes in at 2.13%. How much of that is foundations is difficult to ascertain. Even in CV theory its hard for me to understand many of the titles.

I hope taxpayers are not footing the bill for most of this activity - much of it just intellectual exercises useful for academic promotion, prestige, etc. I've been on grants before and even at the time felt it was not a productive use of public monies. :brow:

Is this analytical geometry? — Daniel

No. Limits don't usually arise in the elementary aspects of the discipline. But boundedness certainly exists for figures like circles and ellipses. However, a parabola is unbounded in the plane. Does a circle in the plane occupy space in the plane? How about a circular disc in the plane?

Analysis at arXiv runs about 25% of the papers in math in May. Foundations 2.13%. Surprisingly, topology runs only 3%. Algebra and its offshoots looks like it dominates.

The American Philosophical Society doesn't seem to separate PhD areas, so its hard to estimate how many are in math.

It's natural that on a philosophy forum discussions of mathematics will mostly lie within the intersection of philosophy and math. I've mentioned in the past that most mathematicians are not heavily involved with foundations, a speculation on my part stemming from my career. It's really hard to pin down exact figures but from the AMS site it would seem that very few degrees from math departments are in foundations, set theory, logic, etc. The 2017-18 stats show that, of 1,960 PhDs, 7% were in a grouping of discrete math, combinatorics, logic, and CS. So that perhaps 3% might have been in foundations, a guess.

Philosophy departments sometimes grant degrees in the philosophy of mathematics, and I don't have any figures there, beyond the figure of 477 total PhDs in philosophy in 2018. Also, CS departments might get deeply into foundations of math. Another glimpse into stats is papers submitted to ArXiv.org, and looking at May it would appear about 2.13% of papers in math were in logic or foundations, out of 1,505 papers submitted. My area(s) - Complex Variables and Dynamic Systems - account for about 8%.

I looked this info up out of curiosity. You may or may not find it of interest. :cool:

This must be related — fishfry

It's notation for an infinite composition like . . . g3(g2(g1(z))). Ultimately the result comes from the structure of functional equations.

which, if I counted correctly, can be expressed in exactly 15 symbols. — fishfry

That probably beats my expansion (for z=1):

$ArcTan(z)=\underset{k=1}{\overset{\infty }{\mathop L}}\,\frac{2z}{1+\sqrt{1+\tfrac{1}{{{4}^{k}}}{{z}^{2}}}}$

You two are going at it, hot an heavy. I'll steal away.

f I understand you correctly, sure, one can feel that one is being religious. I am not disputing that he is accurately describing the way he feels or even views what he does. — TonesInDeepFreeze

I think you are being too literal. I am working on a conjecture right now, as I have on others on and off for fifty years, with a sort of pleasing fervor that could be described more as a religious mood than a purely problem-solving commitment. Its true than mathematicians can be very competitive, even mean spirited at times, and I have been in an unpleasant situation that reeked of it, but exploring math - rather than learning it - is a delight and has its rewards.

so mathematicians will forever be searching for the finite sequence of proofs that are certainly knowable. It's enticing to look for total certainty in your field of study but it can cause problems — Gregory

You are expecting too much of us. We are mere mortals, and few of us commit ourselves to this challenging task.

We can list the essential attributes of religion, and see which of those are attributes of mathematics — TonesInDeepFreeze

No no no. It's what you feel, not what you know. You are thinking like a logician. :gasp:

Some people like to be cheeky like that — TonesInDeepFreeze

Au contraire, he was right!

I don't know what a "set" is, you haven't defined it. — Metaphysician Undercover

In my naivete I once thought of a set as a collection of things called elements. Then I learned the error of my ways. Now I try to avoid thinking of them at all. It's a refreshing experience, like standing at the beach with the soft winds off the ocean caressing your body. :cool:

Mathematics is usually practiced as a Platonic type of religious practice — Gregory

I had a colleague, a fellow math prof, who was a devout Catholic. I once asked him what he thought it was that we do as mathematicians. He said, "We are like priests who practice their faith, it's what we do."

I tried to start a discussion once about what it would mean if all math was wrong and the *opposite* of every equation and theorem was true. — Gregory

That sounds exciting! Sorry I missed it. :lol:

Hmmm. Live and learn. :chin:

Do you think mathematics of the infinite can be done without finite numbers? — Gregory

I don't mess with infinities. Others here do.

. . . but if everything that is real is just infinite sets . . . — Gregory

Whoa! That is quite an assumption. :scream:

Common practice is pretty rigorous once you accept the axioms.

When I gave a mathematical representation of a count. — TonesInDeepFreeze

For your next trick, do one of an earl. :cool:

I'm arguing against accepted mathematical principles — Metaphysician Undercover

How do you feel your campaign is doing?
Has it been worth the struggle?
Have there been casualties?

Are you holding up? :chin:

If you threw something away from you, it would hit you in the back. — noname

In a way this describes this thread. :roll:

What I contend the mathematician is unlikely to do is add the postulate that the probabilities "collapsed" in some sense on the way to the final answer, but rather would just plug in the provided parameters and constraints (i.e. the fundamental constants and whatever values for initial conditions we provide) into the equations and solve for the question. — boethius

That's what I might do. The metaphysics could be fun, though. :cool:

Is it a word game to believe a block of stone has stone outside, and inside, of it as well? Can a thing be hollow and still be a thing? — Don Wade

What of the air inside the garage, inhabiting its space and therefore part of the garage? You open the garage door and the air dissipates as it flows outside the garage. Oh wait, that is part of the garage outside the garage now. As it dissipates further the neighborhood is a former part of your garage. What effect will this have on property values?

These are not just word games — Don Wade

All philosophy arguments are word games. :grin:

This is simply not true. Numbers are defined by quantity, not order. — Metaphysician Undercover

You're failing to distinguish between cardinals and ordinals. — fishfry

MU knows so little of mathematics and yet is so confident. It's almost an admirable trait . . . but not quite.