I was using my knowledge of Jungian and the MMPI which are the most widely used and thus easiest to draw inferences from. — SkyLeach

You have test Briggs-Meyer and MMPI test results of sets of mathematicians that you draw inferences from? How would you obtain such test results?

I'm generalizing and try never to use them to evaluate any individuals — SkyLeach

But you haven't given any basis even for the generalization you stated.

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

If there is one field that is the least based on appeal to authority, I don't know what would be a better candidate than mathematics.

Mathematics is what a consensus of mathematicians says it is. — jgill

I won't argue against the notion that what is the study of mathematics is based on what professionals in departments called 'mathematics' do. But as to whether a purported proof is correct or not (unless it is extraordinarily complicated) is not a matter of consensus. 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.

is an uncountable infinity a mathematical object? — Agent Smith

I don't know of a common definition of 'mathematical object'.

I said that (except in special contexts) in ordinary mathematics, there is no object named by the word 'infinity'. I mean that there is no constant symbol added to the language of ZFC where the constant symbol is rendered in English as 'infinity'.

Instead, there is a predicate symbol that is rendered in English as 'is infinite'. And there are particular constant symbols that are defined as particular sets and we have theorems that those sets are infinite. For example, let 'w' [read as omega] stand for the set of natural numbers. It is a theorem that w is infinite, and it is a theorem that w is countable. Or, let 'R' stand for the set of real numbers. It is a theorem that R is infinite, and it is a theorem that R is uncountable

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.

I meant to ask was whether something uncountable (an example of an uncountable infinity is the set of real numbers R) can be considered mathematical? — Agent Smith

The set of real numbers is defined in set theory, which is a mathematical theory. Set theory makes reference only to pure sets. Not sets of apples or nations or thoughts, but only sets provided by the purely abstract axioms. Indeed, the empty set itself doesn't have to be taken as given but is derivable from purely abstract axioms.

After all, math is, bottom line, about countability (0, 1, 2, 3,...). — Agent Smith

The natural numbers are foundational. That doesn't entail that mathematics must be limited to the natural numbers. And set theory takes not the natural numbers as primitive, but merely the relation 'is a member of', i.e, membership. And with axioms about membership, the natural numbers are constructed, and from the natural numbers, then the integers, then the rationals, the reals are constructed,.

It's still a matter of consensus to determine whether the proof is valid. — jgill

Only when the participants haven't themselves checked the purported proof. They may take the word of the referees that the purported proof is correct. My point is that in principle, it is objective to check whether a purported proof is correct, and if an incorrect inference is clearly shown, then no consensus can alter that the purported proof is not correct. (Again, I'm setting aside situations that are so terribly complicated that there is real debate.)

humans have to agree before it becomes an accepted piece of mathematics — jgill

Granted. But they will ordinarily agree if they check the proof. It's not based on consensus in the same sense as other kinds of questions.

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. — jgill

No, there is a consensus that the purported proof is not correct.

That doesn't entail that mathematics must be limited to the natural numbers. — TonesInDeepFreeze

I understand. :ok:

Yes, it is a mathematical object. — ssu

Counting lies at the heart of mathematics. An uncountable object (e.g. the set of reals), therefore, must be, in some way, nonmathematical, oui?

I see. :up:

Counting lies at the heart of mathematics. An uncountable object (e.g. the set of reals), therefore, must be, in some way, nonmathematical, oui? — Agent Smith

Non.

We must learn to count with the reals, continuous as opposed to discrete, like the naturals.

What do you think of the following problem?

A man invests $7,127 in selling shirts. His selling price is 3 shirts for the price of $200. How many shirts must he sell to break even?

Where n = total sets of 3 shirts each he sells.
The money from selling n sets = 200n

$200n = 7127$

$n = 35.635$

The number of shirts he must sell to break even = $3 \times 35.635 = 106.905$

106.905 shirts!

Well sure. That was why I pointed out the demographic spectrum and how things are changing, but slowly. 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.

In addition, the economic and sociopolitical spectrum tends to limit entry into academics as part of it's statistical weighting towards wealth and privilege.

Finally, the social authority encourages ego which, like the other factors mentioned, retards change and adoption of new ideas.

You didn't ask and even if you did I wouldn't for the reasons I already stated.

To remind you, in case you forgot, it's a waste of my time. Don't care what you infer, you've already demonstrated that isn't going to be affected by anything I say or do.

You didn't ask — SkyLeach

Of course I did. It's on record as a post.

Meanwhile, your characterization of mathematicians, as a generalization, is unsupported by any data whatsoever.

Oh my god! You discovered the hidden truth that there is a rupture in mathematics! Division is not closed in the integers! A discovery as shocking as that Soylent Green is people! And there is not just your example, but thousands of them! Millions of them! Maybe even infinitely many of them! And this contagion is not confined just to mathematics but it affects even the entire garment industry!

↪SkyLeach
I'm somehow not convinced by your words. Uncountability is, in a sense, beyond (conventional) mathematics seen as a counting activity.

True that the variety of numbers has expanded over history. Yet we seem distinctly more comfortable with the category of natural numbers than with any other I can think of. — Agent Smith

Perhaps this will help.
Starting Premises:

• Mathematicians earn a PhD just like all other non-medical disciplines
• Previous is because their doctorate is in a very specific branch of philosophy
• Previous is called the philosophy of Mathematics
• Previous is derived from axioms (self-evident truths)
• The argument to prepend (as opposed to append) ZFC to the philosophy of mathematics is a rational argument based in a self-evident truth, just like all other axioms of mathematics.
• My words are just me talking about that axiom as I understand it.

Look around you (allegorically speaking) and point at anything that is a 1 or a 2 ... or any other number.

Numbers aren't things, they're representations.
Q: What do they represent?
A: Cognitively distinct concepts.
Q: What is a cognitively distinct concept?
A: Anything the rational mind considers a discrete set.
Q: Is there any limit on what the mind considers a distinct concept?
A: Yes, the mind is only able to process allegorical comparisons as set theory and has increasing trouble with concepts sufficiently distant from experience to create one-off pathways (dangling pathways or singular references or [in the case of damage] Disconnection Syndrom
Q: Can you put that in terms I can understand?
A: Maybe. Concepts like massive measurements are very hard for the mind to put into comparative concept and since that's how the mind actually works a great many mistakes are made when doing it until the mind has compensated.
Q: Can you give an example?
A: Yup. Timescales in galactic terms. Distances in interplanetary terms. Infinities. Distance between two points along a curved path like a planet's surface instead of a straight line. Asymmetric periodicity.

Right now, traditionally and with the same consensus as JGill mentioned, numbers are referred to as measurements.

Counting (commutative principle) is just an axiom. Numbers aren't counting except in the sense of combining any two measurements.

When we think of counting apples it can be hard to think it's a measurement. In effect, however, you're measuring volume (just not being very precise). If you're counting apples to fill a pie then the correct answer can vary because the size of the apples definitely will vary based on what kind of apple you are counting. Granny smith vs. red delicious (for example) since granny smith are small green apples and red delicious are quite large red apples.

Ok given that all numbers are just relative measurements between two points in a conceptual context of the mind we can begin to argue for set theory being the first axiom:

• All real numbers are representational measurements of similar sets (i.e. apple = set of apple fruit cells) with a contextually regular periodicity (i.e. each set is defined by the apple skin around it or a complex series of functions that eventually lead back there in a well-understood way).
• Any set can be defined as a vector (a starting point but no limit) - comes from ZFC
• Any set can be defined in terms of its periodicity function (an algorithm, typically a mathematical functioion, that defines the measurement of members of the set) - from ZFC
• Any set defined in terms of its periodicity function is effectively infinite unless bounded by logical rules imposed by the algebraic functions of the definition. (also from ZFC)

I know that's a lot to absorb but it's my best attempt at simplifying the entire argument.

Continuous manifolds cannot be represented by real numbers. A continuous manifold is not made up of points. There are tangent spaces defined on them, and even tangent bundles. From where numbers can be made relating to the curvature of the manifold. You can roll a 2d flat plane over o sphere to return to the initial position. The equivalent would be rolling a ball on an equilateral triangle. You will see that an arrow on the plane will have changed its orientation wrt a drawn equator on the ball, signifying curvature. You can't do this with a cylinder. A cylinder is not curved.What happens in the case of a cone?

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?

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.

Oh my god! You discovered the hidden truth that there is a rupture in mathematics! Division is not closed in the integers! A discovery as shocking as that Soylent Green is people! And there is not just your example, but thousands of them! Millions of them! Maybe even infinitely many of them! And this contagion is not confined just to mathematics but it affects even the entire garment industry! — TonesInDeepFreeze

:lol:

Jokes aside, my example is still in the rational domain. Perhaps I could give an example from construction/engineering: a circular $(\pi)$ dome like the one that tops the Hagia Sophia (Turkey) for an irrational number in the calculation.

Well, you haven't addressed the issue that something uncountable (the set of reals) could be mathematical i.e., to be precise, numerical in nature. The set of reals, as per mathematicians, can't be counted. How can something that can't be counted be mathematical? Can consciousness be counted?

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? — jgill

When is something curved? If the value of the tangent differs from place to place? A circle seems curved. So does a cylinder. So does a torus. You can move a circle through space so it becomes a torus. Maybe the way you move points, lines, surfaces, etc. through the higher dimensional space determines if they are curved (apart from Ricci or Riemann tensors). If I move a point wildly through space the ensuing line gets curvature. A circle hasn't though. A sphere does. But the value of the derivative on it is the same everywhere (how do you define a derivative on a manifold?). A torus has Gaussian curvature but it can be defined such that it has zero curvature. There's more to it...

Can you built a line with points? You can move it through space. Or stack lines to form a surface? Planes to build a volume? What if we take a point from a line? Can you still move on it continuously? Math addresses these questions, and confirms, but still. A line made of points? How you glue them together? How are strings kept on 3d while the 3d soars in 4d?

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 — jgill

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

It's strange that you read those statements as "all sets are" instead of "any set can be". Maybe my word choice was poor? I tend to think in terms of software, not geometry, so there really doesn't need to be a pattern in the numbers in a set just a relationship between them that associates them, even if they're pseudo-random numbers.

Typically a vector is rationally linked with some spatial coordinate system but in software you just can't limit them by guessing the relationship between them from their values.

Not my experience — jgill

This isn't about personal experience or a single academic discipline. When I think of the problems in academia I just don't think about math as one of those. It's not that it isn't one, it's just that the only time the problems ever came up was over the rants against infinity and set theory (which are actually a bit funny to read at times). Mathematics is quite possibly the most empirical of all the sciences.

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.

I double majored but since I graduated I don't think I've read more than a half dozen research papers in mathematics. It's just not one I need to follow closely because new algorithms or solutions sets are rare.

A sphere in 3-D is not composed of points? — jgill

not if you're using set theory. Calc 3 FTMFL

EDIT: damnit I just realized that my perspective will probably get me argued with again...

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.

Nope, I definitely haven't done whatever you said. I also have no clue how to do that because I don't understand.

Nope, I definitely haven't done whatever you said. I also have no clue how to do that because I don't understand. — SkyLeach

Sorry to hear that. Good day.

How can something that can't be counted be mathematical? — Agent Smith

Can a surface be counted? Area can but the surface itself?

How can something that can't be counted be mathematical? Can consciousness be counted? — Agent Smith

Are you arguing this way?:

The set of real numbers can't be counted.
Consciousness can't be counted.
Consciousness is not mathematical.
Therefore, the set of real numbers is not mathematical.

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.