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
"Truth" is negotiable it seems. The word should be avoided in mathematical discussions. — jgill
It just says that true(n) is not a legitimate predicate. — Tarskian
soundness theorem: provable ==> true — Tarskian
So, the correspondentist mapping of truth occurs between theory and "model" (or "universe"). — Tarskian
https://people.math.ethz.ch/~halorenz/4students/Literatur/Semantic.pdf
Soundness Theorem
A logical calculus is called sound, if all what we can prove is valid (i.e., true), which implies that we cannot derive a contradiction. The following theorem shows that First-Order Logic is sound.
https://www.cs.cornell.edu/courses/cs2800/2017fa/lectures/lec38-sound.html
In the last two lectures, we have looked at propositional formulas from two perspectives: truth and provability. Our goal now is to (meta) prove that the two interpretations match each other. We will prove:
Soundness: if something is provable, it is valid. If ⊢φ then ⊨φ.
No, the mapping is from the symbols of the language: — TonesInDeepFreeze
https://en.wikipedia.org/wiki/Correspondence_theory_of_truth
In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world.[1]
Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory attempts to posit a relationship between thoughts or statements on one hand, and things or facts on the other.
https://en.wikipedia.org/wiki/Model_theory
A signature or language is a set of non-logical symbols such that each symbol is either a constant symbol, or a function or relation symbol with a specified arity. A structure is a set M together with interpretations of each of the symbols of the signature as relations and functions on M.
A structure N is said to model a set of first-order sentences T in the given language if each sentence in T is true in N with respect to the interpretation of the signature previously specified for N.
a model is not just a universe — TonesInDeepFreeze
It says that for certain formal interpreted languages, there is no predicate in the language that defines the set of sentences true in the interpretation. — TonesInDeepFreeze
https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
Let L be the language of first-order arithmetic.
Tarski's undefinability theorem: There is no L-formula True(n) that defines T*. That is, there is no L-formula True(n) such that for every L-sentence A , True(g(A)) ⟺ A holds in N.
(in PA or similar)There is no True(n) predicate possible.
model theory will map the symbols. However, the actual purpose of doing that is to achieve what is described in the correspondence theory of truth. — Tarskian
simplify — Tarskian
That seems okay as a broad synopsis. — TonesInDeepFreeze
https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
https://en.wikipedia.org/wiki/Foundations_of_mathematics
This led, near the end of the 19th century, to a series of paradoxical mathematical results that challenged the general confidence in reliability and truth of mathematical results. This has been called the foundational crisis of mathematics. The resolution of this crisis involved the rise of a new mathematical discipline called mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, several parts of computer science.
In principle, mathematics proper is about nothing at all — Tarskian
According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all.
That is extreme formalism. It does not speak for all formalists. — TonesInDeepFreeze
You may hold that the view has merits. I'm only pointing out that formalism is not confined to that view. — TonesInDeepFreeze
Whatever the relative merits, do you see my point that the quote is incorrect, since there are approaches to formalism that don't view mathematics as being about nothing? — TonesInDeepFreeze
“Real” mathematics is almost wholly “useless” whereas useful mathematics is “intolerably dull.”
Whatever the relative merits, do you see my point that the quote is incorrect, since there are approaches to formalism that don't view mathematics as being about nothing?
— TonesInDeepFreeze
Yes, of course. — Tarskian
So why do you quote something that is seriously incorrect? — TonesInDeepFreeze
PA and ZF-inf are two completely different ways of looking at things. "Everything is a natural number" versus "Everything is a set". — Tarskian
Furthermore, besides formalism, there are several other competing ontologies for mathematics. They all turn out to be simultaneously correct as well. For example, Platonism is not wrong either. It is just another way of looking at things. — Tarskian
No. PA can be built from ZF but not the converse. — Lionino
https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-48/issue-4/On-Interpretations-of-Arithmetic-and-Set-Theory/10.1305/ndjfl/1193667707.full
On Interpretations of Arithmetic and Set Theory
Richard Kaye, Tin Lok Wong
2007
This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way.
Impossible, those are two mutually exclusive views. — Lionino
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. — jgill
"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. — jgill
"concept of truth in first order arithmetic statements" — jgill
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. — jgill
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
Isn’t that just an example of Kant’s ‘concepts without percepts are empty? — Wayfarer
Intuition and concepts … constitute the elements of all our cognition, so that neither concepts without intuition corresponding to them in some way nor intuition without concepts can yield a cognition. Thoughts without [intensional] content (Inhalt) are empty (leer), intuitions without concepts are blind (blind). It is, therefore, just as necessary to make the mind’s concepts sensible—that is, to add an object to them in intuition—as to make our intuitions understandable—that is, to bring them under concepts. These two powers, or capacities, cannot exchange their functions. The understanding can intuit nothing, the senses can think nothing. Only from their unification can cognition arise. (A50–51/B74–76)
I found that those of my friends who were admirers of Marx, Freud, and Adler, were
impressed by a number of points common to these theories, and especially by their
apparent explanatory power. These theories appear to be able to explain practically
everything that happened within the fields to which they referred. The study of any
of them seemed to have the effect of an intellectual conversion or revelation, open
your eyes to a new truth hidden from those not yet initiated. Once your eyes were
thus opened you saw confirmed instances everywhere: the world was full of
verifications of the theory. Whatever happened always confirmed it. Thus its truth
appeared manifest; and unbelievers were clearly people who did not want to see the
manifest truth; who refuse to see it, either because it was against their class interest,
or because of their repressions which were still "un-analyzed" and crying aloud for
treatment.
The most characteristic element in this situation seemed to me the incessant stream of
confirmations, of observations which "verified" the theories in question; and this
point was constantly emphasize by their adherents. A Marxist could not open a
newspaper without finding on every page confirming evidence for his interpretation
of history; not only in the news, but also in its presentation — which revealed the
class bias of the paper — and especially of course what the paper did not say. The
Freudian analysts emphasized that their theories were constantly verified by their
"clinical observations." As for Adler, I was much impressed by a personal
experience. Once, in 1919, I reported to him a case which to me did not seem
particularly Adlerian, but which he found no difficulty in analyzing in terms of his
theory of inferiority feelings, Although he had not even seen the child. Slightly
shocked, I asked him how he could be so sure. "Because of my thousandfold
experience," he replied; whereupon I could not help saying: "And with this new case,
I suppose, your experience has become thousand-and-one-fold.
philosophy of the mind is almost never falsifiable — Tarskian
So, what could falsify the thesis you're proposing in this thread? What could someone point to, to demonstrate that your contention 'Mathematical truth is chaotic' is false? — Wayfarer
Not ZF but ZF-inf. It requires removing and denying the axiom of infinity. — Tarskian
Besides, formalism is not an ontology of mathematics, it is an approach to foundations. — Lionino
https://tomrocksmaths.com/2023/10/20/an-introduction-to-maths-and-philosophy-platonism-formalism-and-intuitionism/
Mathematical Formalism is a theory for the ontology of mathematics according to which mathematics is a sort of game of symbols and rules, where new theorems are nothing more than new configurations of said symbols by said rules.
Broadly speaking, Mathematical Platonism (deriving from Plato’s broader theory of ‘forms’) is an ontology of mathematics according to which mathematical objects are abstract, timeless entities existing objectively independent of the circumstances of the physical universe in a separate, abstract realm.
Another crucial tenet of Intuitionist Ontology is a recognition of the temporal nature of our progression of mathematical knowledge over time.
So these are the three big ontologies of mathematics – most other positions, like Empiricism, Psychologism, or Logicism can be more or less categorized as combinations and variants of the primary three.
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.