Not necessarily. The Diagonal paradox can be extended to a sequence of smooth curves that converges to a limit curve in the complex plane in which the disparity of lengths is infinite. — jgill

First: I like the example. I do enjoy coming across these sorts of things.

Second: Mathematicians have a long career of coming across inconsistencies and hurriedly changing the rules so that this particular inconsistency no longer counts.

You may remember that I don't think anything other than relationships exist. I don't think Euclidean space, points, lines or triangles exist.

{Exist = can be described}

The relationships of a triangle do exist. We can describe those relationships.

In the staircase paradox there is a presumption of a continuous manifold within which exist infinitely divisible lines of arbitrary length.

And the definitions of all those terms are circular.

As such, inconsistency in mathematics occurs when people try to describe things that are indescribable.

So long as we only ever try to describe describable things, there is never any inconsistency. But trying to do something impossible always leads to some kind of system error.

We can, then, describe the relationships of our hands. We cannot describe our hands.

We can describe how (the relationships of) our hands pick up (the relationships of) a ball.

The distinction between X and Relationships-of-X is usually irrelevant since we can only ever describe "Relationships of X".

Mathematics tries to describe X. To the extent that descriptions of X and descriptions of X's relationships share common ground this appears to work.

But:

• The Law of Identity defines X as static. X's relationships change.
• Relationships are quantised (there is a minimum measurable change).

This means that the staircase paradox cannot actually define a continuous space or a continuous line. There are no demonstrably continuous relationships for a continuous space to be similar too.

There is no definable limit for a staircase to approach.

P.S.

That isn't very clear.

Simplified:

We can only describe the relationships of X.

Number lines, continuous spaces, points and lines are defined to be static.

It is not possible to describe static (or continuous) objects using dynamic and discrete relationships.

Why can't we just choose to say the set of all sets that do not contains themselves is the highest in order and so is not included in itself? What in math or language requires that include itself in itself? — Gregory

I think Axiomatic Mathematics is wholly mistaken. At the same time, I'm trying not to mislead you about Godel's incompleteness theorems and Axiomatic Mathematics in general.

I think Intuitionism(sp?) is giving up too easily - but it is arguably at least as well founded as axiomatic mathematics (which isn't saying a lot, but...).

Godel is pulling on a piece of string and seeing where it leads: Given a set of assumptions; what conclusions can we draw.

Godel's specific arguments are about what an axiomatic system can say about itself.

As such, the self-referential nature of Godel's arguments is baked into the premise

Godel could have asked different questions - but the questions he asked were specifically and explicitly self-referential. Removing the self-referential bit doesn't leave anything behind - it is all about self-reference.

Axiomatic Mathematics is wrong but the question is right

Axiomatic Mathematics cannot definitively define axioms. Without axioms, Axiomatic mathematics is nothing.

Godel's theorems are part of axiomatic mathematics and fatally flawed even before we reach the self-referential stage.

However, the question of what we are able to say about the universe we inhabit remains a legitimate question.

For example, if we describe electrons as having wavelike properties we are describing one part of the universe in terms of other parts.

Our sense of waves and wavelike properties come from our experience of the universe. We then try to describe the universe by saying that electrons behave (somewhat) like waves.

We are describing the universe as being similar to the universe.

This isn't wrong, of course. We can compare and measure similarities and differences between parts of the universe.

But there is no explanation. We can't step outside the universe and objectively describe it independent of our experience within it.

We are here

All of our experience, understanding and knowledge derives from our existence within the universe.

When we try to describe the universe itself (physics) we find that our measuring sticks are part of the thing we are trying to measure.

We can still make measurements. The Earth's circumference is roughly 40,000km. And a metre is one ten millionth of the distance from the north pole to the equator through Paris.

{The metre is now defined as the distance travelled by the speed of light in vacuum in a particular time.}

As a species, we are getting pretty good at measuring things. But we can't explain what we are measuring.

We don't know what distance is. From small scale to large scale, we can measure distance - but we have no clue how to create it from first principles. We can describe what we observe; but not why we observe in the first place.

Not nihilism

We cannot explain the universe. All of our explanations have the universe as a given.

All we can do is describe/measure one piece of universe using other pieces of universe.

This is a limit on knowledge - if you expect omniscience. We can't explain the universe.

We can describe what the universe is, though.

We cannot say what an electron is. The intrinsic properties of an electron are permanently outside our ability to speculate upon.

We can measure electric fields (whatever they are).

Mathematicians have a long career of coming across inconsistencies and hurriedly changing the rules so that this particular inconsistency no longer counts. — Treatid

In the case of the staircase paradox mathematicians simply accept the fact that the sequence of arc lengths does not converge to the length of the arc that the sequence converges to under the supremum metric on a space of contours. No changing of the rules.

Second: Mathematicians have a long career of coming across inconsistencies and hurriedly changing the rules so that this particular inconsistency no longer counts. — Treatid

Not hurriedly, sometimes it takes centuries.

As a historically contingent activity of humans, math evolves. We extend our concepts to incorporate new situations and paradigms. Negative numbers, complex numbers, transfinite numbers. Set theory, category theory, homotopy type theory.

This is a natural process. You seem to take a pejorative view of mathematical evolution.

I think Axiomatic Mathematics is wholly mistaken. — Treatid

Drat that Euclid, and Ernst Zermelo too.

But if we consider the question in the context of the hyperreal number line - that is, the real number line augmented by adding infinite values at each end, namely Aleph-null and h, the goalposts move. — alan1000

Alan1000, did you get your money's worth from this thread?

.999... = 1 is a theorem of the hyperreals. It must be so, since both the standard reals and the hyperreals are models of the same first-order axioms, therefore they must satisfy the same first-order theorems. This is a simplified statement of a technical fact in model theory.

https://en.wikipedia.org/wiki/Transfer_principle

Aleph-null is not a hyperreal, by the way. You can add symbolic points at plus and minus infinity to the standard real line to get the extended real numbers, but their only use is to make some notation simpler, such as infinite limits and limits at infinity.

https://en.wikipedia.org/wiki/Extended_real_number_line

This is a natural process. You seem to take a pejorative view of mathematical evolution. — fishfry

There is mathematics - and there is the justification/explanation of mathematics.

Applied mathematics is concerned with whether the results are useful. I'm more than fine with this.

It is where pure mathematics tries to establish a foundation of knowledge that I am disgruntled. The effort is laudable - but mathematicians have gotten themselves stuck in a dead end and appear unwilling to extricate themselves.

Axiomatic Mathematics is the show piece of mathematics within which reside logic, formal languages and the majority of mathematical proofs.

But it doesn't work. Axiomatic Mathematics can't define axioms. As deal breakers go - this is one.

There is an argument that a flawed system is better than no system. "We know axiomatic mathematics is flawed - but it is better than nothing".

Except that axiomatic mathematics without axioms isn't merely flawed - it doesn't exist. The axiom bit is not negotiable. You can define axioms or you can't.

As it stands, axiomatic mathematics strives to find the essence of meaning by stripping away extraneous fluff like relationships.

In fact, meaning resides entirely in those relationships.

All progress in modern thought is emphatically despite axiomatic mathematics. The presumption of objective truth has been a catastrophic mistake in modern thought.

There is nothing to be lost by discarding axiomatic mathematics.

As it happens, we can describe relationships. The thing that axiomatic mathematics is trying to dispose of is exactly where knowledge, understanding, meaning and... everything is.

Mathematics' insistence that the path to truth is in defining inherent properties is holding back human progress.

To be fair - mathematics is merely making explicit general societal assumptions. By making implicit assumptions explicit, mathematics makes it much easier to understand what our assumptions are and consider them critically.

I do think that the idea of an objective universe is a dead end and mathematicians should have examined their failures more critically. And we still need the rigour and pedantry of the mathematical process.

No changing of the rules. — jgill

That is debatable.

The central problem is that the rules aren't, and cannot be, defined.

When nobody knows exactly what the rules are, it is hard to determine whether the rules are being followed consistently.

We could spend decades arguing back and forth over whether mathematicians are applying rules consistently to the staircase paradox. And we will never come to a conclusion without a definite and unambiguous statement of what the rules are.

It is impossible to have a definite and unambiguous statement of what the rules are. Definitions are relational. X is not(Everything else). Roughly: A defines B and B defines A.

Any definition of rules (or axioms or anything else) results in infinite regression or circular definitions.

This is where mathematics is stuck. In order to make clear, unambiguous mathematical statements, we first need clear, unambiguous (mathematical) statements.

There is no starting point to jump start mathematics.

..................

I didn't mean to sucker you into a debate on the foundation of mathematics...

But mathematics doesn't have a foundation (c.f. foundational crisis in mathematics).

As much as mathematicians really, really want to make categorical statements - they can't. Anymore than philosophers have managed to establish a fixed point to build philosophy on.

To be clear - I'm not a nihilist. We can and do describe the relationships things have.

It just sp happens that you can either describe X, or the relationships of X but not both.

In this universe - we can describe the relationships of X.

We could spend decades arguing back and forth over whether mathematicians are applying rules consistently to the staircase paradox — Treatid

I don't think you would find a mathematician who would spend more than hour on it.

There is mathematics - and there is the justification/explanation of mathematics.

Applied mathematics is concerned with whether the results are useful. I'm more than fine with this.

It is where pure mathematics tries to establish a foundation of knowledge that I am disgruntled. The effort is laudable - but mathematicians have gotten themselves stuck in a dead end and appear unwilling to extricate themselves.

Axiomatic Mathematics is the show piece of mathematics within which reside logic, formal languages and the majority of mathematical proofs.

But it doesn't work. Axiomatic Mathematics can't define axioms. As deal breakers go - this is one. — Treatid

Ok. You don't like formal math, as you envision it to be. You might be interested to know that there are other approaches to 20th century set theory, such as category theory and homotopy type theory. Some of these areas involve non-traditional intuitionist logic. So you should not think of math foundations as static. Quite a lot of work is being done.

Now I'm not sure if you'e unhappy with axiomatic math, or pure math. As in math for the sake of math, the kind of math that might never be useful. Or it might turn out to be useful in a hundred years or longer.

But you're unhappy with pure math, or math done from axioms. That's ok with me. I am not the Lord High Defender of Math. If you don't like pure math that's ok with me.

But having said that I'm perfectly ok with whatever your feelings are in this matter, let me try to address some of your concerns the best I can.

There is an argument that a flawed system is better than no system. "We know axiomatic mathematics is flawed - but it is better than nothing". — Treatid

Better than nothing for what? You know, if you ask a thousand pure mathematicians what axiom system they're using, or if they can state the ZFC axioms, they'd look at you funny. They don't study axioms. They study the math of quantum field theory, or exotic topological spaces, or deep properties of the natural numbers. They're never thinking about foundations. They have no opinions on foundations and they wouldn't even understand the question. It doesn't come up.

Just ask @jgill, who has had a career as a professional mathematician without having much contact with the foundational side of things.

It's like living in a house. How often do you climb underneath the house to play around with the foundation? Most people just hire specialists for that.

So nobody is making any compromises with anything or using a flawed system. They're just doing math.

Except that axiomatic mathematics without axioms isn't merely flawed - it doesn't exist. The axiom bit is not negotiable. You can define axioms or you can't.

As it stands, axiomatic mathematics strives to find the essence of meaning by stripping away extraneous fluff like relationships.

In fact, meaning resides entirely in those relationships. — Treatid

Aha. I can answer that.

It's like the game of chess. It has formal rules. Within the game there are moves and positions that are legal; and moves and positions that are not legal, according to the rules. The rules are more or less arbitrary. None of it "means" anything; but people themselves invest the game with varying levels of meaning, from the occasional player to the tournament grandmaster. People devote their lives to the game, yet the game has no intrinsic meaning at all.

If it helps, you can think of math that way. It's a formal game. It means nothing. Some people enjoy it. Some make it their life's work.

If some physicist, or architect, or bank teller, comes along and has some use for math; all the better. It makes no difference to the pure mathematicians; except insofar as it motivates their universities to employ them to teach some low-level math skills to the physicists and bank tellers; and then leave them alone to do their obscure, pure research.

So what's wrong with that?

All progress in modern thought is emphatically despite axiomatic mathematics. The presumption of objective truth has been a catastrophic mistake in modern thought. — Treatid

Of all the intellectual disciplines, math is the least concerned with objective truth; and makes no claims to it whatsoever. A mathematician will tell you that proposition P implies proposition Q within some particular axiomatic system; but they never claim it means anything about the real world.

You should take up your complaint with the physicists and bank tellers. They're the ones trying to apply all of this fictitious math.

There is nothing to be lost by discarding axiomatic mathematics. — Treatid

You want to make all those math professors go out and get real jobs? That's cruel.

But what do you mean? There would be nothing to be lost by throwing out music, or mountain climbing, or chess, or sports. None of those are necessary to life, they just make life worth living. Entertainments. Fields of study.

People study the 15th century British kings and queens. They study the great wars. They read literature or pulp novels. Why on earth should't those who are so inclined, study abstract math?

And besides -- once in a while, a previously useless area of math becomes useful. Mathematicians invented non-Euclidean geometry in the 1840's. It was regarded as a curiosity. Then in 1915 Einstein found that it was just the thing for him to express his new theory of relativity.

So math is often about reality, but in the future. And if you ban all the pure math, you'll lose all those tools that might be needed.

As it happens, we can describe relationships. The thing that axiomatic mathematics is trying to dispose of is exactly where knowledge, understanding, meaning and... everything is. — Treatid

If you like the math of relationships, you should chec out category theory. It's the "abstract theory of arrows." You have a bunch of things and a bunch of arrows between pairs of things. Relationships if you like. You can express quite a lot of math in that framework.

https://en.wikipedia.org/wiki/Category_theory

Mathematics' insistence that the path to truth is in defining inherent properties is holding back human progress. — Treatid

You are just making that up. It's your own straw man. There are no mathematicians asserting any such thing. Mostly they're just trying to crank out the next paper to justify their latest grant; and teaching calculus to the budding physicists and engineers.

Where are you getting such ideas that mathematicians are insisting on any such thing?

To be fair - mathematics is merely making explicit general societal assumptions. By making implicit assumptions explicit, mathematics makes it much easier to understand what our assumptions are and consider them critically. — Treatid

Math doesn't make any assumptions about society. Are you thinking about quantitative sociology perhaps? Or epidemiology, where we apply statistical methods to see how diseases spread

Mathematicians don't do the things you think they do.

If I may be so bold as to tell you: All of this is entirely in your head. Your ideas do not refer to anything real about mathematics. You are tilting at windmills of your own imagination.

I do think that the idea of an objective universe is a dead end and mathematicians should have examined their failures more critically. And we still need the rigour and pedantry of the mathematical process. — Treatid

By the time I got to the end of this I did not think you had an argument or thesis at all. Speculations regarding the nature of the universe belong to the philosophers and sometimes the cosmologists and quantum physicists.

Definitely not mathematicians. When mathematicians study quantum field theory, they do the math. They don't do the metaphysics. And if they do, at that moment they are acting as philosophers and not mathematicians.

You have your basic facts all wrong.

You have your basic facts all wrong. — fishfry

:clap: :cool:

You have your basic facts all wrong. — fishfry

You put too much effort in a post towards someone who won't learn from disagreement. In another thread, he said that he is quite sure about he was talking about, and in the same post he said that the twin paradox is a paradox in Newtonian physics only and not in relativity.

People study the 15th century British kings and queens — fishfry

Study who?

You put too much effort in a post towards someone who won't learn from disagreement. In another thread, he said that he is quite sure about he was talking about, and in the same post he said that the twin paradox is a paradox in Newtonian physics only and not in relativity.

I had to read through that individual's entire post to realize it didn't make any sense. No matter. Maybe something I wrote was interesting to someone.
— Lionino

People study the 15th century British kings and queens
— fishfry

Study who? — Lionino

Sorry did I get my history wrong? Not sure what you meant. I was making the point that people find value in studying all kinds of stuff so why not math. Was my analogy off the mark?

Your quote includes something I didn't say. So I am not sure what you are replying to.

Oh ok. I get it now.

You put too much effort in a post towards someone who won't learn from disagreement. In another thread, he said that he is quite sure about he was talking about, and in the same post he said that the twin paradox is a paradox in Newtonian physics only and not in relativity. — Lionino

I had to read through that individual's entire post to realize it didn't make any sense. No matter. Maybe something I wrote was interesting to someone. — fishfry

.

Sorry did I get my history wrong? Not sure what you meant. I was making the point that people find value in studying all kinds of stuff so why not math. Was my analogy off the mark? — fishfry

No, it was a contrarian joke implying that people (me) don't study them.

https://en.wikipedia.org/wiki/Category_theory — fishfry

Speaking of category theory, I came in contact with it (again) to explore the subject of vector spaces with irrational dimensions. Naturally, vector spaces traditionally defined have a dimension n, n E N, naturally because the set of its basis can't have π elements, but something like that is the case of fusion categories, if a mathoverflow user is to be trusted.

I had not heard of fusion category, but found it has 7 views per day on Wikipedia. And Categories has a little over two papers per day on arXiv.org . These numbers give a very crude estimate of a subject's popularity. If I were a lot younger and healthier I might try to learn something of this topic. Never thought of vector spaces with irrational dimensions either.

Your quote includes something I didn't say. So I am not sure what you are replying to. — Lionino

Sorry, must be quoting issues.

No, it was a contrarian joke implying that people (me) don't study them. — Lionino

Oh I get it.

Speaking of category theory, I came in contact with it (again) to explore the subject of vector spaces with irrational dimensions. Naturally, vector spaces traditionally defined have a dimension n, n E N, naturally because the set of its basis can't have π elements, but something like that is the case of fusion categories, if a mathoverflow user is to be trusted. — Lionino

Wow that's a new one on me. The Wiki page on the subject is brief and unhelpful.

How would a vector space dimension work that wasn't a positive integer? Have a link?

ps ... found this.

https://math.stackexchange.com/questions/1466820/vector-spaces-with-fractional-dimension

There is this link https://ncatlab.org/nlab/show/fusion+category
It mostly goes over my head

found this. — fishfry

That is the stack user I was referring to.

, Thanks. But way, way outside the scope of vector spaces I ever encountered, like spaces of contours in the complex plane (where I still dwell).

But way, way outside the scope of vector spaces I ever encountered, — jgill

Pretty weird stuff ...

It is where pure mathematics tries to establish a foundation of knowledge that I am disgruntled. The effort is laudable - but mathematicians have gotten themselves stuck in a dead end and appear unwilling to extricate themselves. — Treatid

It is not a popular function of "pure" mathematics to delve into these issues. For example, arXiv.org lists the number of math papers submitted in this last week: Total-783, Logic-5, History&Overview-1. To compare: Category Theory-18, Complex variables-18. Others-751 (29 additional categories).