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.

Trump was already president for four years and he didn't end democracy. — fishfry

Some tend to conveniently forget that. Trump made an attempt to control the border, then when Biden came into office he made that infamous comment, "storm the border". And don't forget the Afghanistan debacle.

You are not old as Godel's proof — fishfry

Not quite. The mathematicians I knew BITD had little to no interest in discussing the distinctions between provability and truth. We were mostly in classical (complex) analysis. Mostly we are gone now. A few of us remain.

the true nature of the truth — Tarskian

Run that by me again, please.

This is not more than one order, it is just different aspects of one order — Metaphysician Undercover

Deep stuff, here. :roll:

and the order would be the three balls. Right? — javi2541997

Seems like a peculiar use of the word "order".

Et tu? ChatGPT doesn't know anything about mathematical philosophy. It just statistically autocompletes strings it's been fed. — fishfry

Here is a quote from Reddit that brings some clarity to the subject of "truth" in mathematics these days:

Godel's completeness theorem, applied to group theory, says that any statement that's true for every group can be proved from the axioms of group theory. Similarly, there is more than one model of ZFC. The existence of various models of ZFC is analogous to the existence of different groups. Some statements are true in one model of ZFC and false in another. Such a statement is independent of ZFC.

I'm an antique. Truth for me is associated with proof.

I'm done here. Sorry. Maybe another mathematician will appear.

if someone else is the candidate, it's a wild card, things could shift very quickly. — Wayfarer

True. All the polls up to that point mean very little. The whole environment changes.

If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order — Metaphysician Undercover

Set consisting of three balls colored red, white and blue. They also have differing weights. What is THE order? Just curious.

I was suggesting that a slowing down according to a convergent series might count as stopped, since it would never reach the limit or "0". — Ludwig V

Some time ago I mentioned time dilation in relativity theory in this regard.

That needs work — TonesInDeepFreeze

It is a tad simplistic. But it is as far as I went in that direction in my career; as for infinity, I never quite reached it for it lay beyond bounds. It's good you and fishfry are more up to date. Thanks for your service.

Why not go directly into 2D. If you stay away from the SB-tree and Niqui arithmetic I might linger a bit longer. Let's see. You might even tempt fishfry back.

First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets — ssu

Called the New Math in the USA. I can't even imagine this in grade one. I taught elements of it in college algebra courses in the 1970s - but not for long.

These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry

Here is what ChatGpt has to say about mathematical truth:

In mathematics, truth is typically understood within the framework of logical consistency and proof. Here are a few key aspects of truth in mathematics:

Logical Consistency: Mathematical statements and propositions must be internally consistent. This means that there should be no contradictions within a mathematical system. For example, in Euclidean geometry, the parallel postulate is consistent with other axioms, but in non-Euclidean geometries, different parallel postulates lead to different but internally consistent geometries.

Verification through Proof: In mathematics, a statement is considered true if it has been proven using rigorous logical arguments based on accepted axioms and definitions. The process of proving involves demonstrating that the statement follows logically from these axioms and previously proven statements (lemmas).

Objective Reality: Mathematical truth is independent of human beliefs or opinions. Once a mathematical statement has been proven, it is universally accepted as true within the mathematical community. This aspect of objectivity distinguishes mathematical truth from truths in other domains, which may depend on subjective interpretation or observation.

Unambiguity: Mathematical statements are precise and unambiguous. Each term used in mathematics is defined rigorously, and the rules of inference and logical operations are well-defined. This clarity ensures that the truth of mathematical statements can be objectively assessed.

Scope of Truth: In mathematics, truths are often considered to be eternal and immutable once proven. For example, the Pythagorean theorem, once proven, remains true indefinitely and universally applicable within the domain of Euclidean geometry.

In essence, truth in mathematics is grounded in rigorous logical reasoning, proof, and adherence to accepted axioms and definitions. It is a fundamental concept that underpins the entire discipline, allowing mathematicians to build upon previously established truths to explore new areas and make further discoveries.

Do you care play a more active role in the discussion or would you rather leave it at that and let this thread 'dry up and vanish'? — keystone

Some time back this thread shifted to the idea of starting math with continua and deriving points, rather than the other way around. MU has spoken of this, but has yet to put any meat on the bones. You, on the other hand, got into the discussion with some sort of ideas, and I was intrigued. I assumed you might begin with something akin to contours in the plane, but you went another direction, and sticking with one dimension I think was very limited, and rather boring I fear.

There are many thousands of ideas, large and very small, floating around in the world of mathematics these days, each one championed by one or more individuals. I enjoy playing with contours in the complex plane, and I hoped what you had to say would somehow involve this concept. But, instead, the discussion moved towards reconstructing the reals, devolving into an obscure approach - interesting I am sure to a few - but not to the, relatively speaking, many.

If you were to return to the beginning and speculate continua that precede points, or something similar, the thread might continue. Just my opinion.

Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry

Me too.

I no longer have any idea what we are conversing about — fishfry

Perhaps the paper by Milad Niqui. In that case things may get technical and out of the realm of TPF.

viz,

What I'm proposing is that Niqui arithmetic is more fundamental than the SB-tree. — keystone

Does the axiom of identity mean Ludwig V = keystone ?

Just curious. :smile:

HALFTIME COMMENTARY: For those viewers who might wonder if this thread analogizes everyday discussions in mathematics among its various practitioners, let me assure it does not - at least from my antiquated perspective. Expertise in the "finer" points of logic is rarely required in traditional math, although,I admit, I've lost track of the enormous varieties of the subject over the passing years.

And perhaps I am wrong: checking ArXiv.org I see that in the past week there have been around 25 new logic papers submitted - about the same number as those in my area, complex analysis. And the axiom of extentionality on Wikipedia garners about 60 views per day - a healthy enough following.

Just passing thoughts when reflecting on the current discussion. Kudos to the three or four involved. :clap:

but all of these relations stem from Niqui arithmetic. — keystone

Unknown territory for me. No Wikipedia page I can find (among 26,000+), but perhaps it's under a different heading. You are full of surprises. Are you Niqui? :cool:

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. — keystone

Each row of the tree involves medians, which require ratios of integers and arithmetic of these ratios. So, your top down approach always involves bottom up procedures. You cannot correlate rational numbers with nodes without using expressions like a/b. Instead of simplifying, you are complicating something you assume. Just my opinion.

I'm a computational fluid dynamics analyst — keystone

I've wondered about that. Thanks for illuminating.

Sounds like your professor just didn't like foundations — fishfry

You might think so from what I said, but he was young and pretty enthusiastic about teaching the subject. We had numerous worksheets that eventually led to the construction of the exponential function. So, his comment at the end came as a bit of a surprise. :cool:

I agree with this sentiment. Whether it's noncomputable reals, the halting problem, Gödel's incompleteness theorems, or the liar's paradox, they are all screaming at us that there is a potential in mathematics that cannot be fully actualized. But Classical mathematics aims to actualize everything — keystone

Not true. I published papers when I was active that never assumed infinity was actualized. Fryfish and I, sometime back, argued about the use of transfinite math in analysis, particularly functional analysis. He pointed to the use of Zorn's lemma or the axiom of choice as a required tool to prove the Hahn-Banach theorem, and I replied that that was true, but by altering the hypotheses slightly, they were not required. Hahn-Banach was my only very brief encounter with transfinites in my career. But then I sought interesting theory in classical analysis - a far cry from foundations. So, your statement is not entirely correct.

But I admire your tenacity.

Well I didn't become a mathematician! I got to grad school and my eyes glazed. — fishfry

In my first semester as a grad math student I was required to take a course called Introduction to Graduate Mathematics. It was basically naive set theory and at the end the professor said, "You should only continue in foundations if this course really appeals to you. How many of you find that to be the case?" I recall out of thirty students one or two hands went up. The rest of us wiped the glaze out of our eyes and went on into other areas of mathematics.

But this was 1962, back in the dark ages. And at a state university, not a top-notch school, like Harvard. Things have changed since then. When were you in a grad math program?

Three spatial dimensions, one time dimension. Spacetime. Don't try to make time into a fictitious part of space. But who really cares?

Should I be talking about a bijection between the non-dimensional points on a line and the set of integers? — Ludwig V

You can use that term, but only if you are more specific about "points on a line" and specify natural numbers or rational numbers corresponding to these points. That's it.

So if I had said "And when we describe the principle of distinction between non-dimensional points on a line, we find that our counting with natural numbers is endless", you would have agreed? — Ludwig V

No. If "the points on a line" correspond to integers or rational numbers, yes. Way too vague.

Real numbers are uncountable. — jgill

I see. Why can't I count with natural numbers? — Ludwig V

"principle of distinction between non-dimensional points on a line" does not specifically speak of natural numbers. Language play.