• jgill
    3.7k
    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.
  • Tarskian
    301
    "Truth" is negotiable it seems. The word should be avoided in mathematical discussions.jgill

    Well, no. The term "truth" should be used in a way that is compatible with its model-theoretical definition, which is in fact not particularly negotiable.

    In model theory, truth is a correspondentist notion.

    A fact is true because it is part of a particular collection of truth, i.e. a "model", an "interpretation" -- or if the operations supported are irrelevant-- a "universe".

    If such "model" "interprets" a theory, then every statement that is provable from this theory will be true in the "model", i.e. soundness theorem:

    soundness theorem: provable ==> true

    So, the correspondentist mapping of truth occurs between theory and "model" (or "universe").

    Concerning Tarski's undefinability, it doesn't say that truth does not exist. It just says that true(n) is not a legitimate predicate.
  • TonesInDeepFreeze
    2.8k
    It just says that true(n) is not a legitimate predicate.Tarskian

    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.

    soundness theorem: provable ==> trueTarskian

    That's not the soundness theorem.

    The soundness theorem stated in two equivalent ways:

    If a set of sentences G proves a sentence S, then every model of G is a model of S.

    If a set of sentences G proves a sentence S, then for all models M, if every member of G is true in M then S is true in M.

    So, the correspondentist mapping of truth occurs between theory and "model" (or "universe").Tarskian

    No, the mapping is from the symbols of the language:

    each individual constant map to a member of the universe

    each n-place predicate symbol maps to an n-ary relation on the universe

    each n-place operation symbol maps to an n-place function on the unviverse

    And ""model" (or "universe")" is wrong since a model is not just a universe. Rather, for every model there is a universe for that model.
  • Tarskian
    301
    Agreed. However, the more precise the definition is being phrased, the more impenetrable the explanation tends to be. Therefore, in a multidisciplinary context, it may be preferable to just give the gist of the explanation.

    So, for example, "soundness means: provable => true" is just the gist of it. In fact, it seems to be ok to phrase it like that:

    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

    That is how it is technically achieved. I was trying to point out that it achieves the same goal as stated in the correspondence theory of truth:

    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.

    Technically, 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.

    The following explanation about correspondence in model theory will probably be deemed impenetrable in a multidisciplinary context:

    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.

    But then again, in my opinion, the correspondence theory of truth is perfectly fine to describe the gist of how model theory sees the relationship between theory and model.

    a model is not just a universeTonesInDeepFreeze

    Well, I glossed over that, without insisting too much. If you give the technical explanation of what exactly is missing, then pretty much nobody will keep reading in a multidisciplinary context.

    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

    I simplified the following:

    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.

    To:

    There is no True(n) predicate possible.
    (in PA or similar)

    Again, the complete statement above is probably too much in a multidisciplinary environment. It will be deemed impenetrable.
  • TonesInDeepFreeze
    2.8k
    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

    That seems okay as a broad synopsis.

    simplifyTarskian

    Simplifications are okay if they don't mislead by omitting crucial conditions and distinctions.
  • Tarskian
    301
    That seems okay as a broad synopsis.TonesInDeepFreeze

    There are surprising and unexpected connections between the foundational crisis in mathematics and fundamental metaphysics.

    In principle, mathematics proper is about nothing at all:

    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.

    If you dig into the foundational crisis of mathematics, however, it suddenly starts talking about deep metaphysical issues. The mathematical crisis arose out of profound paradoxes:

    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.

    How can something that is essentially about nothing at all, suddenly make a U-turn, and give answers on the fundamental nature of everything?

    The mathematical crisis turns out to have massive implications for the following issues in metaphysics:

    - What is truth?
    - What is the connection between a truth-bearer and a truthmaker?
    - What is free will?
    - Is the universe predetermined?
    - Is the universe part of a larger multiverse?
    - Is there a heaven and a hell?

    The mathematical crisis even puts into question the most fundamental and seemingly unassailable laws of logic:

    - What is actually "identity", since the law of identity does not always hold?
    - Why is the law of the excluded middle (LEM) not always legitimate?
    - Since identity and the LEM are in question, is even the law of noncontradiction actually circumstantial?

    The mathematical crisis also shows that existing answers in metaphysics are largely unsatisfactory:

    - It shines another light on Kant's Critique of Pure Reason. Kant has probably got it mostly wrong.
    - It certainly proves the positivists wrong.

    After more than a century, the implications of the mathematical crisis have not been digested in metaphysics. There is pretty much no awareness of its metaphysical impact. In my opinion, this is because people in both fields almost never talk to each other. One reason for this, is the fact that most publications on the mathematical crisis are written in a language impenetrable to outsiders.
  • TonesInDeepFreeze
    2.8k
    In principle, mathematics proper is about nothing at allTarskian

    That is one extreme view.

    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.
  • Tarskian
    301
    That is extreme formalism. It does not speak for all formalists.TonesInDeepFreeze

    It is actually an incredibly productive view. The more you insist that it is about nothing at all, the more it starts revealing secrets about everything. It is truly mind blowing.
  • TonesInDeepFreeze
    2.8k


    You may hold that the view has merits. I'm only pointing out that formalism is not confined to that view.
  • Tarskian
    301
    You may hold that the view has merits. I'm only pointing out that formalism is not confined to that view.TonesInDeepFreeze

    The reason why moderate formalism has less merit that the most extreme take on the matter, is actually to be expected. If you abstract away almost everything, then the very little that is still left, will indeed apply to pretty much everything.
  • TonesInDeepFreeze
    2.8k
    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?
  • Tarskian
    301
    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. Hardy famously said:

    “Real” mathematics is almost wholly “useless” whereas useful mathematics is “intolerably dull.”

    I would add to what Hardy said, that "useful" mathematics has absolutely zero metaphysical implications. That is why it is "intolerably dull".

    Mathematics proper is indeed not necessarily about nothing. That is why it is so boring.
  • TonesInDeepFreeze
    2.8k
    I find mathematics to be the opposite of boring.
  • TonesInDeepFreeze
    2.8k
    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?
  • Tarskian
    301
    So why do you quote something that is seriously incorrect?TonesInDeepFreeze

    The quote is extreme.

    I don't think, however, that it is incorrect.

    If mathematics is "just string manipulation" then it is indeed "about nothing". As I have already acknowledged, other views on the matter are also viable.

    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.

    Similarly, concerning competing mathematical theories, PA and ZF-inf are two completely different ways of looking at things. "Everything is a natural number" versus "Everything is a set".

    However, they turn out to be perfectly bi-interpretable.

    You can express natural numbers as sets, and arithmetic on natural numbers as set operations, and then everything you say about natural numbers, you can effectively say them about these sets. The reverse works fine as well.

    Extreme formalism turns out to be a metaphysically useful view.
  • Lionino
    2.1k
    PA and ZF-inf are two completely different ways of looking at things. "Everything is a natural number" versus "Everything is a set".Tarskian

    No. PA can be built from ZF but not the converse.

    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

    Impossible, those are two mutually exclusive views.
  • Tarskian
    301
    No. PA can be built from ZF but not the converse.Lionino

    Not ZF but ZF-inf. It requires removing and denying the axiom of infinity.

    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, they are mutually exclusive. Still, they both provide a perfectly legitimate ontology for mathematics. Similarly, you can build a society on capitalism or on communism. They are both mutually exclusive.
  • fishfry
    3.2k
    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

    That was not my point. Mathematicians don't use the word true in their formal work.

    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." On the contrary, I imaging that you say, "I learned something about nonabelian widgets" or whatever. Am I wrong? I would be surprised if I'm wrong?

    "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

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

    "concept of truth in first order arithmetic statements"jgill

    Yes but it's so unlike you to have an interest in Tarski's undefinability theorem.

    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

    The "philosophical" discussions in this forum do not reflect the actual work of philosophically inclined mathematicians. For example the early category theorists like Mac Lane were very philosophically oriented. "He was vice president of the National Academy of Sciences and the American Philosophical Society, and president of the American Mathematical Society ..." Impressive resume.

    https://en.wikipedia.org/wiki/Saunders_Mac_Lane

    It's a bit like physicists. Most of them are of the "shut up and calculate" school, while others -- a minority -- are interested in what it all means, what it can tell us about the ultimate nature of reality. They call the latter foundations of physics. So math is to math foundations as physics foundations. Most don't care, some do.
  • jgill
    3.7k
    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".)
  • Wayfarer
    21.4k
    If mathematics is "just string manipulation" then it is indeed "about nothing".Tarskian

    Isn’t that just an example of Kant’s dictum ‘concepts without percepts are empty’?
  • Tarskian
    301
    Isn’t that just an example of Kant’s ‘concepts without percepts are empty?Wayfarer

    When Kant writes about philosophy of the mind:

    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 generally refuse to engage, if only, because philosophy of the mind is almost never falsifiable. That is why I ignore a good part of the text in "Critique der reinen Vernunft". If what Kant says, is simply not actionable, I will just generously concede the point to him. What else can I do?

    In this regard, about similar theories, Karl Popper writes in "Science as falsification":

    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.

    My opinion about "Critique of Pure Reason" is pretty much the same as what Popper writes about Marx, Freud, and Adler. The number of falsifiable points in what Kant writes, is very, very limited. Still, when Kant -- very rarely -- takes the risk of saying something that is actually falsifiable, it always turns out to be false.
  • Wayfarer
    21.4k
    philosophy of the mind is almost never falsifiableTarskian

    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? Isn't Popper's point that metaphysical theses cannot be disconfirmed by empirical discoveries? What empirical discovery would disprove the thesis 'Mathematical truth is chaotic'?
  • Tarskian
    301
    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

    If you demonstrate that Cantor's theorem is false. (the existence of countable and uncountable infinity)
    If you demonstrate that Gödel's theorem is false. (incompleteness)
    If you demonstrate that Tarski's theorem is false. (undefinability of the truth)
    If you demonstrate that Turing's theorem is false. (halting problem)
    If you demonstrate that Carnap's theorem is false. (diagonal lemma)
    and so on.

    These theorems are all interrelated. Demonstrate one flaw in one of their proofs. One is probably enough, because one flawed theorem will be enough grounds to demonstrate the falsity of all other ones.

    In another paper, Yanofsky argues that it is Cantor's theorem that is at the core of it all:

    https://arxiv.org/pdf/math/0305282

    "You cannot create an onto mapping between a set and its power set."

    Until I ran into Yanofsky's other paper, I used to think that Carnap's theorem was the real culprit:

    "For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both."

    My intuition says that Yanofsky is probably right, and that it is Cantor's theorem that is at the root of it all, but I am currently still struggling with the details of what he writes.

    In the paper about the chaos in the truth about the natural numbers, Yanofsky argues that:

    If you try to express all the truth about the natural numbers, you are effectively trying to create an onto mapping between the natural numbers and its power set, the real numbers, in violation of Cantor's theorem. That is why most of this truth is simply ineffable, and a fortiori, unprovable, and therefore, unpredictable.
  • Wayfarer
    21.4k
    All well beyond my competence I’m afraid. I have a vague recollection of a BBC documentary, Dangerous Knowledge, ‘Documentary about four of the most brilliant mathematicians of all time, Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing, their genius, their tragic madness and their ultimate suicides.’
  • Tarskian
    301

    Concerning Cantor, Gödel, and Turing, I have some kind of morbid fascination for what I consider to be a form of disaster tourism.

    It is akin to a guided tour around Chernobyl reactor number four. It leads to the very fault lines in the tectonic plates in the foundation of things, which are indeed fiendishly ugly.

    Sometimes, it is even difficult to believe that it is all true. It often gives the sensation that "it cannot be that bad?".

    Sometimes, I don't really get it, or not immediately. At that point, I know that I am close to understanding something that is even worse than all the bad stuff that I have come across already.

    I cannot stop because I like too much playing with metaphysical fire. If you have the sensation that you are about to discover the true secret name of Satan, would you stop or would you keep going?
  • Lionino
    2.1k
    Not ZF but ZF-inf. It requires removing and denying the axiom of infinity.Tarskian

    That was a typing mistake obviously as the very post I quoted said "ZF-inf" instead of just "ZF". Regardless, ZF-inf and PA are not "two completely different ways". One is tied to the other.

    Two things that are mutually exclusive cannot "turn out to be simultaneously correct as well". It is absurd. Besides, formalism is not an ontology of mathematics, it is an approach to foundations.
  • Tarskian
    301
    Besides, formalism is not an ontology of mathematics, it is an approach to foundations.Lionino

    Apparently, other people call formalism also an ontology:

    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.

    Platonism and intuitionism are in his opinion the other main ontologies:

    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.
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