Your comments give me pause to consider the use of the expression "intentional ambiguity" may be inappropriate for what I had in mind. The statements (assuming no division by 0)
$F=\frac{tK}{a-\left( a-\frac{tK}{r} \right)}$, $b=a-\frac{tK}{r}$ provide information about b - something called a bifurcation - but when reduced to lowest terms using very simple algebra all that is left is $F=r$, which is also relevant information about F. A kind of double entendre I suppose, or maybe something much more elementary. Your suggestions are appreciated.

I'm also curious why you just skipped over the whole part about a vecor being defined by element association since that was the mathematics perspective and the practical application perspective. — SkyLeach

These are vectors as I know them. I guess I don't understand what exactly you are talking about. That could be me. But if you point out your description in the article that would be good.

how do you define a derivative on a manifold? — EugeneW

differentiable manifolds

Set theory isn't just sets of points, it can also be a scene described as a space (hilbert, sobolev, etc...) with objects described functionally instead of sets of points. I deal far more with scenes described rather than sets of points except when rendering a solution set. — SkyLeach

Your perspective of "set theory" is not the normal math perspective. If it works for you, fine.

When I talk about many of the problems in academia I tend to be thinking of cosmology, astronomy, paleontology, the humanities (psych, anthro, socio, etc...) The more empirical and rigid a discipline is the less they seem to get into academic problems — SkyLeach

OK. Not a topic I have an opinion about.

Your example was the Necker cube. As I quoted from Wikipedia the two possible interpretations exclude each other by way of contradiction — Metaphysician Undercover

If one expects two pieces of information simultaneously, yes. But with a slight passage of time one perceives cube#1, then a moment later cube#2. Two different objects arising from one symbol.

Any set can be defined as a vector — SkyLeach

The sequence <2,7,9> can be seen as a vector. The set {2,7,9} cannot, since the positions of the elements is arbitrary.

There is too much direct control asserted over too much of each generation's career by the previous generation, causing the normal evolution of thought and culture to be retarded in academics — SkyLeach

Not my experience at all. After my PhD and getting a tenured position I belonged to a small international group of academic mathematicians, all of whom eagerly sought new ideas, novel ways of looking at things, unusual results, etc.

What you might be referring to is when a grad student has to choose a project (dissertation) and is not capable of making that choice, their advisor will guide them onto a path he thinks they are able to follow and hopefully do original research - the fundamental requirement for the degree. And, sad to say, sometimes the advisor will in effect do much or most of the original research and give credit to the student.

However, it is true that some departments are "governed" by a cliche that exerts pressure to push ahead toward certain research goals, having little patience with deviations. I've seen this also. This can occur when the cliche forms around a prominent, celebrated academic. Especially when there are grants to be captured.

Any set can be defined in terms of its periodicity function — SkyLeach

What's that? So a set of random integers is defined that way? Do you speak of a set or a sequence?

Lots of material here for TIDF.

Continuous manifolds cannot be represented by real numbers. A continuous manifold is not made up of points. — EugeneW

Do you mean differentiable manifolds? A cylinder created by moving a circle through space is not curved? A sphere in 3-D is not composed of points?

rather than your claim that contradictory interpretations could be simultaneously correct — Metaphysician Undercover

Complementary, not contradictory.

But as to whether a purported proof is correct or not (unless it is extraordinarily complicated) is not a matter of consensus — TonesInDeepFreeze

It's still a matter of consensus to determine whether the proof is valid. Yes, in some abstract realm a proof is valid or not according to logical principles, but humans have to agree before it becomes an accepted piece of mathematics. Occasionally a proof is so long and so complicated the verification process is difficult.

Whatever consensus there might be, if one shows an incorrect inference in a purported proof, then the proof is disqualified from being deemed an actual proof. — TonesInDeepFreeze

Guess there's not a consensus, then.

Academics is, at its core, an appeal to authority. — SkyLeach

In a sense. An appeal to the authority of consensus. — jgill

↪jgill I don't really know what you meant by that. — SkyLeach

Mathematics is what a consensus of mathematicians says it is. That's the "authority".

According to Wikipedia, the Necker cube is an ambiguous drawing, "it can be interpreted to have either the lower-left or the upper-right square as its front side". My argument is that neither of the two possible interpretations is the correct one. — Metaphysician Undercover

Why do you keep bringing up the word correct? The only thing correct is one gets two pieces of information from one image, like my mathematical example (a bit too complicated to relate here) - one expression yields two pieces of math information depending on how it is interpreted (seen).

This topic has run its course for me. You can have the last word. :roll:

Pure math, and all other forms of signification, once uncoupled from empirical experience, become unintelligible. — ucarr

Each day recently arXiv.org has received about 200 research papers in math. Many of these are "uncoupled from empirical experience", yet thousands of math people find them intelligible. However, the general public will not.

Numbers, uncoupled from interrelated material objects, become random, unable to signify anything intelligible. — ucarr

A good calculus student might disagree.

Pure math, and all other forms of signification, once uncoupled from empirical experience, become unintelligible.

Numbers, uncoupled from interrelated material objects, become random, unable to signify anything intelligible. — ucarr

These are very rigid statements that are beliefs, not facts. You should indicate as such. Should a philosopher state their beliefs as facts?

Academics is, at its core, an appeal to authority. — SkyLeach

In a sense. An appeal to the authority of consensus.

Being valid does not necessarily imply "correct", because the conclusion must also be sound.In the case of meaning, the true meaning is the one intended by the author, that is what is meant. — Metaphysician Undercover

Therefore we can conclude that the true meaning is that neither is the correct one. — Metaphysician Undercover

In my case the true meaning is a dual observation: giving one piece of information when viewing from one perspective, and another when viewing from the other perspective. Take a Necker cube for example. It can be seen two ways, each a valid cube. What is "the meaning intended by the author"?

Therefore we must conclude that in the case of intentional ambiguity neither is the correct interpretation. — Metaphysician Undercover

Your conclusion does not logically follow. I have a mathematical expression that can be interpreted two distinct ways, each of which is valid and "correct". However, it is a novel idea and something I haven't seen in math before. Maybe I'm wrong? Who knows . . .

I quoted myself because you seemed to have missed my detailed comparison when you asked where Quaternions came into things. — SkyLeach

Were (are) you an aeronautical engineer or Naval person? I know little of rotation theory, but the gimbal lock problem can occur in one of these schemes but not the other. The short paper I linked is as close to this as I get. I almost always work in the complex plane.

I was joking about the behavior of fellow mathematicians. In all my days I never saw a rant. And your description of a math person's personality is valid sometimes, but more often they are social animals - the practice of mathematics is a very social activity. I recall being at an autumn meeting at the Luminy campus of the University of Marseilles in 1989 at which there was communal dining and quite a jovial atmosphere. And a summer meeting at the University of Trondheim in 1997 where a member of the royal family attended a convivial banquet overlooking a ski jump where their Olympic team put on a performance. Other international meetings displayed similar atmospheres.

Sorry about the ramble above, but many assume math talent means the sort of personality you have described. Which, incidentally, omitted the fact that many in the profession have musical talent.

I would argue that intentional ambiguity results in neither one being correct — Metaphysician Undercover

Intentional ambiguity is the use of language or images to suggest more than one meaning at the same time

(Cambridge English Dictionary)

But in the case of simple maths, it's impossible to disagree that the sum of two and two is four, obviously (although I have an ominous feeling..... :scream: ) — Wayfarer

How could you!!!!! :worry:

Well, that's an interesting approach to mathematics. I'm always curious about how laymen interpret the subject. :cool:

Returning to mathematics; the purpose of mathematics is to validate the properties and functions applied to the sets. It makes no assertions, descriptions or assumptions about the nature of the sets. It's intended to strictly regulate validation of function and derived outtcome only — SkyLeach

You refer repeatedly to "sets". Care to expand on that ?

Pass two people the same proposition or axiom and have them each explain it. They will not explain it with the exact same expressions. Therefore they do not have the same interpretation. It's a very simple and obvious fact which you seem to be in denial of. — Metaphysician Undercover

The fact they may explain it using slightly different words does not imply they interpret a theorem differently. Even if they do there is a specific interpretation that is correct.

Of course, I am thinking of theorems I have created (or discovered) that have simple mathematical ideas most mathematicians would agree upon. If you asked me to explain symplectic geometry after a brief exposure to the subject I would surely botch it up. That doesn't mean my interpretation is on some kind of par with an accurate appraisal of SG.

This discussion concerns the obvious: yes, we may interpret differently. No, all interpretations are not correct according to some recognized authority. But it leads to a more challenging notion: intentional ambiguities, like neckers cube. And I recently posted a short note concerning a math expression that implies two distinct conclusions depending on how one interprets it. Both interpretations are correct simultaneously.

Is there a mathematical and or logical expression for comparing the properties and lack of properties of Objects? — Josh Alfred

Way too vague. Weight, size, shape, etc. involve different mathematical expressions. Although fundamental particles have properties, spin, momentum, etc. that can be collectively described in tensors or matrices or ??

This thing needs to go to the Lounge.

Not sure where set theory comes into play here. Quaternions are more algebraic and geometric. But I will admit the part of calculus I least enjoyed teaching is spacial coordinates and rotations and translations. And the closest I've come to exploring Euler angles and quaternions is in SU(2):Dynamics of LFTs of SU(2)

Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be — SkyLeach

I can write very concise and precise definitions and descriptions, then use them in sheer babble.

As I said, I believe a theorem is literally the terms that state it. Therefore any and all theories or theorems are open to individual interpretation. Each of us understands them according to one's own experience of learning and practicing. — Metaphysician Undercover

Not true. I may open a math paper on a topic with which I am unfamiliar and guess at what it is about, but this in no way opens the theorems therein in any professional sense to individual interpretations. Yes, I may interpret them the wrong way, just as you have on countless occasions misinterpreted the simplest of mathematical symbolism. If I were to insist it was my right to reinterpret results I would be ridiculed for my stance - as I should be.

Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything. — SkyLeach

It's interesting you have come upon a well-kept secret within the mathematics community. I, myself, have witnessed such rants at the top echelons of the profession. It's an embarrassing spectacle and rather unexpected within a discipline so rigorous and demanding calm reasoning. Sad. :cry:

So if "the theory" exists within the rational mind, manifested as the activity which is "understanding", then we cannot accurately call it "the theory" any more, because each person has one's own unique interpretation of what is called "the theory", so we would have a multitude of different instances of the same theory. — Metaphysician Undercover

Not so. This "theory" is composed of a number of specific theorems not open to individual interpretation. But the "meaning" of this theory certainly is an individual's prerogative.

A more philosophical "theory" might fit your description.

Infinity broke math — Agent Smith

Necessary to allow the goodness to pour forth. :cool:

Your example is greater in quantity not in quality.

Big difference. — Joe Mello

Which perhaps you might have clarified in your OP. Since this is more a religious discussion I'll bow out.

We can measure things and we can make linear statements about those things with mathematics but beyond that we have to use linear algebra — SkyLeach

Lots of non-linear material out there. That's where the study of differential equations gets interesting. So much more.

Generally, "exist" is a spatial-temporal concept. To exist is to be describable in spatial-temporal terms — Metaphysician Undercover

I'm not so sure. Does a theory of infinite compositions of complex functions exist? Why yes, it does exist. I should know. Lots of other examples.

Welcome to the forum, junior :cool: It needs more youngsters like you!

No combination of lesser things can create a greater thing without something greater than the greater thing added to the lesser things. — Joe Mello

The union of the sets {a,b,c} and {d,e} is the set {a,b,c,d,e} which is "greater" than the initial sets. I'll not comment on the more general statement.

I haven't read the book [Penrose], but from reviews I get the impression that his Mathematical Reality is essentially the same thing as Virtual Reality. If that is not questioning our traditional understanding of reality (Materialism & Atomism) I don't know what it's all about. :cool: — Gnomon

Could be. I'm only commenting on parts of his gigantic book I've read. Mostly it is what you would expect of a mathematical physicist: lots of physics math, some of which is like traditional math, some beyond my pay grade. Math physicists mostly search for math to predict the results the experimentalists are getting. But some dabble in philosophy of reality. Some conjure up woo.

Let me toss another one on the fire :razz: : Stephen Wolfram's A New Kind of Science. I'm not advocating, just mentioning — Real Gone Cat

I have that book, also. It too is gigantic, over 1,000 pages, and devoted to Wolfram's mathematical approach to physical reality: elementary cellular automata. I've read parts, but again I doubt that more than a few have read the entire tome (that used to be a cocktail party joke). I wrote several programs in BASIC to produce the automata he suggests in the complex plane, and got interesting imagery. But as a fundamental concept explaining physical reality his ideas have failed.

In recent years, several scientists have questioned our traditional understanding of Reality, both intuitive and academic. Here's just a few, writing in the last 25 years. Theoretical physicist Carlo Rovelli : Reality Is Not What It Seems (quantum reality); mathematical physicist Roger Penrose : Road to Reality (quantum ideality) — Gnomon

I have Penrose's book (2004) and have read portions over the last few years. The subtitle is "A Complete Guide to the Laws of the Universe" , and I don't find him in general questioning our traditional understanding of reality, only trying to explain how we have reached such understandings and how math has been indispensable in this process. However, his discussions of quantum effects becomes more speculative.

The book is immense, over 1,000 pages, so it's doubtful very many readers have read it all.

If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?

I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.

The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future. — sime

We seem to be looking at a common notion from different perspectives. I assume you are a CS person (?). First of all a student in a math class is more likely to ask, What is an infinite sequence?, whereupon the definition as a function of the positive integers is provided. Then comes the distinction between a sequence in which each term is defined by a formula and one that self-generates by recursion. Examples would make these ideas clear:

(1) ${{S}_{n}}=S(n)=3n+1,\text{ }n\in N$

(2) ${{S}_{n}}={{S}_{n-1}}+{{S}_{n-2}},\text{ }{{S}_{1}}=1,{{S}_{2}}=1,\text{ }n\gt2$

Then the idea of a finite expansion (largest value of n) or an infinite expansion (non-terminating).


$\lt{{S}_{1}},{{S}_{2}},\cdots ,{{S}_{n}}\gt\text{ , }\lt{{S}_{1}},{{S}_{2}},\cdots \gt$


Then, of course, there are special and/or more complicated cases, like a sequence which is non-terminating but assumes the same value after a certain point, or a sequence that is convergent in a metric space, etc.

In all my years I don't recall using the expression "potential infinity".

What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. — sime

I think you may have misinterpreted the Wikipedia article:

potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps

There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition.

You have a very sophisticated writing style that I find a bit hard to process, but others may not. And when a person writes well there is a temptation to put aside statements that might give one pause. I'm guilty, but fortunately TonesInDeepFreeze is sharper in this regard.

If maths are proven by showing the effectiveness, then it is correlated to empiricism — javi2541997

Within mathematics proofs are the product of reasoning. However, in physics a mathematical process may show effectiveness without a reasoned proof. Feynman did this with his path integral. When this happens mathematicians try to catch up and provide a foundation by rigorous proof. Sometimes they do and sometimes they don't. There are anomalies in regularization, for example.

3. Unknown unknowns ( :zip: )

Each category of unknowns might deserve separate treatment — Agent Smith

I find 3. particularly appealing. What might be the first step along this path?

Gödel’s own incompleteness theorems proving the limited and unprovable nature of all mathematical endeavors — Photios

Hmm :chin:

“If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” — What is Math?

What nonsense.

You can look it up if you really want to know more (it might make your head hurt). — Real Gone Cat

Thanks. That's enough.