I hardly understand anything in this thread, as my knowledge of mathematics is rudimentary. — Wayfarer
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
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
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
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
In your work, do you think of yourself as discovering formal derivations? Or learning about nonabelian widgits? — fishfry
For example the early category theorists like Mac Lane were very philosophically oriented. — fishfry
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
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
What made you quote that? — fishfry
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'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
I know that Russell wanted to develop math from logic, and Gödel busted Russel's dreams. Beyond that I am totally ignorant — fishfry
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
logic being a niche, ignored by most math departments — fishfry
Trump was already president for four years and he didn't end democracy. — fishfry
You are not old as Godel's proof — fishfry
the true nature of the truth — Tarskian
This is not more than one order, it is just different aspects of one order — Metaphysician Undercover
and the order would be the three balls. Right? — javi2541997
Et tu? ChatGPT doesn't know anything about mathematical philosophy. It just statistically autocompletes strings it's been fed. — fishfry
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.
if someone else is the candidate, it's a wild card, things could shift very quickly. — Wayfarer
If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order — Metaphysician Undercover
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
That needs work — TonesInDeepFreeze
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
These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry
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
Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry
but all of these relations stem from Niqui arithmetic. — keystone
Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. — keystone
I'm a computational fluid dynamics analyst — keystone
Sounds like your professor just didn't like foundations — fishfry
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
Well I didn't become a mathematician! I got to grad school and my eyes glazed. — fishfry
Should I be talking about a bijection between the non-dimensional points on a line and the set of integers? — Ludwig V
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
Real numbers are uncountable. — jgill
I see. Why can't I count with natural numbers? — Ludwig V
There can be no counting to begin with. — jgill
I'm surprised. Could you explain why? — Ludwig V