I think that the vast majority of academic papers are considered to be irrelevant. In that sense, it does not matter if the justification supplied is solid or not. Nobody cares anyway — Tarskian

Pretty much the case in mathematics. One result is that even competent referees skim over details too often, especially if the author is a respected academic. Lots of mistakes are published, mostly non critical.

The mathematics of this is precise. A fractal distribution system has a log/log or powerlaw scale of size. That is how a geography can be efficiently covered so every drop of water or wannabe flyer gets an equal chance of participating in a well-organised network of flow — apokrisis

Interesting reference. A parabolic fractional distribution is a little obscure with the Wiki page getting only 7 views per day.

Thanks for the info. :cool:

The nearby city of Pueblo (111,000) was given a government grant to re designate several two way streets into one way with a bike lane. For the past six weeks I drove these streets to medical appointments each weekday, and saw a total of three bike riders.

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).

Interesting comments. For me, much stems from density of life. My daughter lives in Brooklyn in an area having roughly 36,000 inhabitants per square mile, while I live on the high prairie of southern Colorado in a ruburban environment of about 400 per square mile. The necessity of private transportation to some extent is a function of this statistic.

Years ago I was on a train going from London to Wales, passing through lovely meadow lands, when I saw a large vertical building rising abruptly in the distance, surrounded by a high wall. An apartment building made to cram many into a relatively small space. All around were fields empty with the exception of a few cows and trees.

High density or low density. A largely political argument. I saw a cartoon recently that showed differences between Democrats and Republicans, and for housing the Dems favored the high rise building, cramming as many as possible together regardless of ethnicity or religion or political persuasion. The Republican model was a cottage in a yard filled with grass and flowers with a cute white fence surrounding it.

That is why I have personally never treated and will never treat philosophy or mathematics as more than just hobbies — Tarskian

Understandable. Many others likewise. For the last 24 years math explorations have been a hobby for me as well.

Mathematics does not have direct practical applications, mostly by design so. That is often a good thing, but it also means that the academic consensus has much more weight than it would have, if there were practical applications — Tarskian

There's just the mutually back-patting consensus, or else meaningless grades on a collection of otherwise irrelevant tests and exams, or even the eternally back-patting citation carousel. That is why I have personally never treated and will never treat philosophy or mathematics as more than just hobbies — Tarskian

Your opinion has been noted. Actually, I agree with the first statement above. The second sounds a little bitter.

, 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).

Not sure if mathematical logic is just a curiosity — Tarskian

To make your point would require some sort of poll of mathematicians asking "Are the Foundations of Mathematics important to you as you pursue your explorations into your specialties?"

I'm betting most of my colleagues would say no. I think you are possibly unaware of the enormous scope of mathematical inquiries these days. Look up college catalogues of math courses and find how many have set theory prominently displayed in more than an introductory course. Here are Harvard's Offerings. M145a and M145b and M385 out of how many courses? Plus, of course some overlaps.

When I was somewhat active over 25 years ago Foundations never came up at the conferences I participated in - international groups. Except a joke or two about the continuum hypothesis.

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.

One could say the crisis is still going on, as we don't know whether ZFC is free of contradictions (and perhaps never will). — Lionino

I suppose it is, especially among foundations mathematicians. But I would not say it remains a crisis within the broader scope of the profession. Mostly a curiosity.

You have your basic facts all wrong. — fishfry

:clap: :cool:

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.

To me, "shake on it" signifies agreeing to a course of action rather than agreeing on a belief. If you meet someone and ask if they too believe in Kamala Harris and they say yes, do you then shake on it? I would not.

(1eg) If a theory explains an observation, then the theory is evidenced. — Hallucinogen

What is that flying across the sky leaving behind it a trail? It must be Icarus on his way. Yes, my theory is evidenced! :roll:

I actually did not invent the term "foundational crisis of mathematics" by myself — Tarskian

OK. Thanks for the links. It should be emphasized that the crises is in the philosophy of mathematics. Mathematicians by and large ignore the crises (unless they are into fundamentals). :cool:

Here we go again, assuming a stroll along an uneven path is the same as wandering through a minefield. — jgill


So, the idea is that the use of Godel numbering in a logic expression points to making use of the philosophical capability of the language and therefore turns the expression into a philosophical one. There may be exceptions, though. — Tarskian

I was just commenting on your referring to "a foundational crises in mathematics". I doubt many mathematicians would agree there is a "crises". Concerns perhaps.

The origin for what I write, is of course, the foundational crisis in mathematics — Tarskian

Here we go again, assuming a stroll along an uneven path is the same as wandering through a minefield.

How is all this relevant for defining philosophy? How is this the relevant to philosophy in any way? — Ludwig V

:up:

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.

Mathematics has a massive foundational crisis with insurmountable issues. — Tarskian

That is why I find the foundational crisis in mathematics an exhilarating subject — Tarskian

Well, it's good someone is interested. :roll:

Nothingness was abhorrent ? Geometry? Don't know and don't care. :cool:

Math originally came from accounting, believe it or not — frank

Well, here is what ChatGPT has to say:

Mathematics and accounting are deeply intertwined, but mathematics did not originally come from accounting. Instead, mathematics has a much broader and older origin that spans various domains.

Here’s a brief overview of how these fields are related:

Early Mathematics: The origins of mathematics date back to ancient civilizations such as the Babylonians, Egyptians, and Greeks. Early mathematics involved basic counting, measurements, and arithmetic. These practices were crucial for various practical activities like agriculture, trade, and construction.

Accounting Origins: The practice of accounting, especially systematic bookkeeping, has roots in ancient civilizations as well. For instance, the Sumerians developed one of the earliest known accounting systems around 3000 BCE, which involved recording transactions on clay tablets. Accounting was essential for managing resources, trade, and taxation.

Development of Mathematics: Mathematics evolved from these practical needs into a more abstract and systematic study. Ancient Greeks, such as Pythagoras, Euclid, and Archimedes, made significant contributions to mathematics that went beyond mere accounting and measurements, exploring geometry, number theory, and more.

Interconnection: As mathematics developed, it increasingly influenced and was influenced by accounting practices. For example, the development of algebra and calculus provided tools for more sophisticated financial analysis and modeling.

In summary, while accounting and mathematics are closely related and have influenced each other, mathematics as a discipline predates accounting and encompasses a much broader range of study than accounting alone.

In mathematics - a paradox (inconsistency) demonstrates a faulty set of axioms — Treatid

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. There is no argument I have heard of that implies fundamental axioms of the real (and complex) numbers is at fault. I seem to recall Aristotle was aware of this discrepancy of lengths.

Is there a question here? There is a lot more to science than honest bookkeeping.

edit: didn't see the previous replies.

When I donate to Wikipedia, in a sense I receive a benefit indirectly and help provide benefits to others. In a way altruistic philosophy I suppose. When I transfer money from one savings account to another at the same interest rate its as though nothing has happened Unless the second account has an additional name on it.

Describe why arbitrary transfers are philosophically significant — Mark Nyquist

Donations can shape society. Simply moving money around usually does not.

This thread is a stretch. :roll:

Abraham Robinson's definition of h revolutionised mathematics in the 1960's. — alan1000

This is an exaggeration. There are probably universities around where this is taught regularly, but it has not caught on to any significant degree in general. A colleague of mine who taught at the U of Colorado told me they made an attempt to start a course in the subject, but it flopped. I don't see any course in their curriculum now that focuses on non-standard analysis. But there are courses in foundations where it may crop up.

So, rather than drift off into systems that depart from the standard material on the real numbers, its best to stick with the widely accepted ideas. Just my opinion.

Virtually every professional mathematician lives in the world created by this movement. Nobody notices because it's like fish not noticing water. — fishfry

Thus, were set theory removed mathematicians would perish. I think not. But mathematics would not be nearly as robust as it is today. My humble opinion.

Back in the late 1960s my advisor remarked on the separation of the nitty gritty at ground level and the efforts to fly high and look down on mathematics, an abstract perspective to see how the various parts fitted together and document how parts from one branch were like parts form another. He gave me a choice and I felt far more comfortable working in the lowlands, (particularly after learning a bit about algebraic topology). I came into the profession exploring convergence and divergence of analytic continued fractions and related material. Pretty much an extension of the efforts during the 1700s and 1800s to solidify those properties of series. Grubby stuff, but I still enjoy grovelling in it. :cool:

Even worse than Wikipedia, which much too often is, at best, slop. Quora is close to the absolute lowest grade of discussion. It is a gutter of misinformation, disinformation, confusion and ignorance. Quora is just disgusting — TonesInDeepFreeze

Another discussion: Are mathematical articles on wikipedia reliable?

"[...] as an introduction to a topic Wikipedia is very good."
I'll fix that: as an introduction to a topic Wikipedia is very good lousy. — TonesInDeepFreeze

Two opposing opinions. Here is a discussion on Quora.

Wikipedia articles about mathematics are too often incorrect, inaccurate, poorly organized or poorly edited — TonesInDeepFreeze

There are over 20,000 articles about math on Wikipedia. My own experience has been that accuracy improves with advanced topics, and I have found that as an introduction to a topic Wikipedia is very good. But I know little of foundations.

It's call the arithmetization of analysis. It's a thing in late 19th century math. Basically founding math, including calculus and continuous processes, on set theory.

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

5 views per day. The title doesn't resonate with many apparently (including me). Nevertheless, an important movement.

My favorite game on the internet is guessing the number of page views per day for math and other topics. I guessed 126 here, whereas it is 111. Close, but no cigar. — jgill

That's interesting. Which page views? I think you've mentioned in the past that you look at papers written or something like that. — fishfry

Daily pageview statistics on Wikipedia. And papers submitted to ArXiv.org

For example: True arithmetic (Talk)
And True arithmetic (pageviews)

Low priority in Mathematics in Wikipedia. About the same as my own low priority math article.

(The daily analysis can be misleading, however. The median is a better indicator of popularity. For example, I just checked my former sport, bouldering, and found a huge disparity with a daily average of 912, but a median of 351. It was running below 400 per day until one day only it shot up to nearly 12,000. I haven't a clue.)

So truth and falsity, semantic concepts, are always relative to a particular model. The integers and the integers mod 5 both satisfy the same ring axioms, but 1 +1 + 1 + 1 + 1 = 0 is false in the integers; and true in the integers mod 5.

That's what we mean by truth. Mathematical truth is always:

Axioms plus an interpretation. — fishfry

Thank you. This is similar to the group theory example. It makes more sense now.

Mathematicians are starting to use https://en.wikipedia.org/wiki/Proof_assistant , proof assistants and proof formalizer software. It's a big field, going on ten or twenty years now — fishfry

My favorite game on the internet is guessing the number of page views per day for math and other topics. I guessed 126 here, whereas it is 111. Close, but no cigar.

I don't think Jan 6th happens unless Trump gives the speech he gave right before — RogueAI

Possibly an act of sedition, but not one of treason.

Wikipedia and article:

A sentence in the language of first-order arithmetic is said to be true in N if it is true in the structure just defined.

A sentence in the language of first-order arithmetic is said to be true in N {\displaystyle {\mathcal {N}}} if it is true in the structure just defined

It looks like passing the buck to me. The word "true" in mathematics appears to be a kind of primitive when used outside of "true by virtue of proof". However, the statement of Goldbach"s Conjecture from Wikipedia:

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

might very well be true in the common sense of the word, even if possibly unprovable. But one cannot actually assert it is true - only that it might be.

Hopefully, Biden will be eased out of the race and replaced by a more worthy opponent for Trump. Kamala Harris is good at reading teleprompters, but does she have presence of mind and ability to argue off the cuff?

Interesting to hear you arguing against the concept of truth — fishfry

Ordinarily I would not give it much thought, but this thread seems to focus on math truth beyond virtue of proof. You seem to know what that is all about. Can you provide a very simple definition of this sort of truth in math? I suppose the definition of a triangle is truth without proof. Truth by definition. But what makes a string of symbols true? Model theory? I thought I understood a parallel idea when I quoted the group theory example from StackExchange, but I guess not. Are axioms true by virtue of definitions?

I have a friend who is a math PhD. I have never really had a chance to discuss this sort of thing in depth, but I have asked him before if he though mathematics was something created or discovered. He said "created" but not with any great deal of confidence and waffled on that a bit — Count Timothy von Icarus

I'm guessing, typical. Philosophical speculations distract from True mathematics. :cool:

You don't believe in the word truth, or that anything in the world is true, even outside of math? — fishfry

True or False?: The Earth is a planet. Answer: True (by virtue of classification)

True or False?: The square of the hypotenuse in a right triangle equals the sum of the squares of the two sides. Answer: True (by virtue of proof)

True or False?: The Continuum Hypothesis is true. Answer: Well, let's see . . . .

Did you think your work was "about" anything? Or pure symbol-pushing?
I'm pressing you on this point because I don't believe you did not believe in the things you were studying! — fishfry

I never spent any time thinking about what I was doing. I did it, and still do it because it is a fascinating realm of exploration. As was rock climbing when I was a lot younger. I never puzzled over the fundamental nature of mathematics. And I doubt my colleagues did either.

Gravity is true, wouldn't you say? — fishfry

No. Gravity simply is. Some aspects could be said to be true. Word babble IMO.