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.

A well-known mathematician takes a look at "truth" in mathematics:

Desparately Seeking Mathematical Truth

I hardly understand anything in this thread, as my knowledge of mathematics is rudimentary. — Wayfarer

I'm with you. And I was a professor of mathematics. I am still puzzled over what precisely "true" means beyond verification by formal proof.

If you point to a number, '7', what you're indicating is a symbol, whereas the number itself is an intellectual act. And furthermore, it is an intellectual act which is the same for all who can count. It's a very simple point, but I think it has profound implications — Wayfarer

I like your clarity.

What of us who think it is both created and discovered? — jgill

Sounds contradictory to me, unless you are saying the application of it is discovered — Lionino

It is a bit fuzzy. But here is an example: Linear fractional transformations have been around for years but at some point someone discovered they could be categorized by the behavior of their fixed points. One such categorization was "parabolic", in which the fixed point demonstrates both attracting and repelling behaviors. Thus, a category was both discovered and created. When I determined the conditions under which infinite compositions of parabolic transformations converge to their fixed points years ago that was a discovery based upon a creation.

And speaking of which, category theory could be considered a creation, then its characteristics follow as discoveries.

However, I am open to other perspectives. Most mathematicians don't care to argue the point. But it is certainly fair game for the philosophically inclined.

Derivative problem. If you are a platonist, you think math is invented, if you are a nominalist or conceptualist, you think math is discovered. — Lionino

What of us who think it is both created and discovered?

But when you make a discovery, don't you feel that you are discovering something that is true, or factual, about whatever it is you're studying? Surely you don't lean back and say, "That's a cool formal derivation that means nothing." — fishfry

Nor do I lean back and say, Wow, that's true! I simply don't use the words "true" or "truth" when doing math. I don't even think the words. But that's me, not other math people.

In your work, do you think of yourself as discovering formal derivations? Or learning about nonabelian widgits? — fishfry

I don't think of myself doing anything. I only do. Or did. I'm pretty old and not in such great shape to do much of anything.

For example the early category theorists like Mac Lane were very philosophically oriented. — fishfry

Doesn't surprise me. I am (was) a humble classical analysis drone, far from more modern and more abstract topics. Maybe young math profs these days use the word "truth" frequently.

(On the other hand I did point out what I considered the truth of a form of rock climbing many years ago by demonstrating and encouraging a more athletic, gymnastic perception of the sport. Even then I didn't use the word "truth".)

There is nothing wrong with referring to truth in mathematics. (1) The everyday sense of 'truth' doesn't hurt even in mathematics. When we assert 'P' we assert 'P is true' or 'it is the case that P'. — TonesInDeepFreeze

Of course there is nothing wrong with using the word "true" in math. But in the papers I have written (around thirty publications and over sixty more as recreation) I doubt that I ever used the word - but I could be wrong. On the other hand, "therefore" is ubiquitous.

I assume you think of your research as discovering truths about abstract mathematical structures that have some Platonic existence in the conceptual realm. You surely feel that the things you study are true. Do you not? — fishfry

"True but verify" might be my motto. I suppose I would consider myself a Platonist were I to care, but this type of philosophical categorization - although relevant to this forum - matters very little to me.

What made you quote that? — fishfry

"concept of truth in first order arithmetic statements"

If there are any practicing or retired mathematicians reading these threads I wish you would speak up. I would ask my old colleagues what they think of these philosophical discussions, but they are pretty much all gone to greener pastures.

Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. . . . Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics. — Metaphysician Undercover

It is true that some mathematicians are "problem solvers", perhaps the majority. But for the others, myself included, a mathematician is an explorer trying to find a path extending knowledge in a particular direction or discovering new directions. Creation and discovering are two sides of the same coin: we create, for instance, simply by virtue of defining and we discover where those creations lead.

It's been obvious from the outset that Trump projects all the evils he commits onto his enemies. What is really depressing is the ease with which it is believed, even by some here — Wayfarer

True indeed. Many here do believe that Trump is guilty of this.

I know that Russell wanted to develop math from logic, and Gödel busted Russel's dreams. Beyond that I am totally ignorant — fishfry

I agree. "Truth" is negotiable it seems. The word should be avoided in mathematical discussions.

Tarski's Undefinability Theorem says (Wiki):

Informally, the theorem says that the concept of truth of first-order arithmetic statements cannot be defined by a formula in first-order arithmetic. This implies a major limitation on the scope of "self-representation". It is possible to define a formula T r u e ( n ) whose extension is T ∗ , but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic. However, this formula would only be able to define a truth predicate for formulas in the original language L. To define a truth predicate for the metalanguage would require a still higher metametalanguage, and so on

Biden took a major hit with the debate and Trump scored a major victory with the ear bullet. Trump's side is energized awaiting his VP pick and Biden's is in a scramble trying to convince him to throw in the towel. — Hanover

Well said.

I think logic is concentrated in a few places but not that widely. Seems that way anyway. — fishfry

I just checked on this past week's papers in logic posted at ArXiv.org . Four are from American universities and 13 are from foreign countries. FWIW

logic being a niche, ignored by most math departments — fishfry

Depending upon the quality of the university to some extent. With the exception of a 12 month post-graduate program I took at the U of Chicago for the USAF, my entire education was in large state universities (4).

I checked at what Harvard has to offer and they have two undergraduate courses in mathematical logic (and probably foundations), but at my last Alma Mater there is nothing of that kind offered at any level.