Except it doesn’t allow for the iunreasonable effectiveness of mathematics in the natural sciences. — Wayfarer

As I said above, the reason the most people won’t defend platonism is because they don’t understand or can’t live with the metaphysical commitment it entails. Myself, I have no such difficulty. — Wayfarer

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

Pythagoras, in his teachings focused on the significance of numerology, he believed that numbers themselves explained the true nature of the Universe. Numbers were in the Greek world of Pythagoras' days natural numbers – that is positive integers (there was no zero).

Any observations on the arguments for or against mathematical platonism as outlined in this post? — Wayfarer

https://en.m.wikipedia.org/wiki/Philosophy_of_mathematics

Davis and Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.

Kurt Gödel's Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly.

So you don't accept that 7=7? — Wayfarer

https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html

In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.

R. Kossak and J. H. Schmerl, The structure of models of Peano arithmetic, vol. 50.

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.

That is one single nominalist out of the several nominalists. Stop misrepresenting the view that you found out about by reading a Redditpedia article two hours ago.
And your criticism is spurious. — Lionino

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

Leśniewski denied the existence of the empty set and held that any singleton was identical to the individual inside it.

The principle of extensionality in set theory assures us that any matching pair of curly braces enclosing one or more instances of the same individuals denote the same set. Hence {a, b}, {b, a}, {a, b, a, b} are all the same set. For Goodman and other proponents of mathematical nominalism,[30] {a, b} is also identical to {a, {b} }, {b, {a, b} }, and any combination of matching curly braces and one or more instances of a and b, as long as a and b are names of individuals and not of collections of individuals.

In the foundations of mathematics, nominalism has come to mean doing mathematics without assuming that sets in the mathematical sense exist. In practice, this means that quantified variables may range over universes of numbers, points, primitive ordered pairs, and other abstract ontological primitives, but not over sets whose members are such individuals. Only a small fraction of the corpus of modern mathematics can be rederived in a nominalistic fashion.

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

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.

"Penrose argues that human consciousness is non-algorithmic, and thus is not capable of being modeled by a conventional Turing machine, which includes a digital computer." — fishfry

IOW the owners of oracle could just tell it to lie to Thwarter. — Bylaw

https://en.m.wikipedia.org/wiki/Lambda_calculus

There is no algorithm that takes as input any two lambda expressions and outputs TRUE or FALSE depending on whether one expression reduces to the other. More precisely, no computable function can decide the question. This was historically the first problem for which undecidability could be proven.

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

So why do you quote something that is seriously incorrect? — 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.”

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

That is extreme formalism. It does not speak for all formalists. — TonesInDeepFreeze

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.

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.

There is no True(n) predicate possible.

I try to keep an open mind and take the good with the bad of all, say, a bit eccentric posters. I hope that is not too uncharitable to Tarskian. Am I being fair? — fishfry

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.

But given that, my original point stands. That programs can't have free will. And I hope you agree that humans being deterministic would not contradict that point. — fishfry

"Truth" is negotiable it seems. The word should be avoided in mathematical discussions. — jgill

Yet this is also the general issue that Yanofsky is talking about as this is found on all of these theorems. — ssu

Spirituality is an intellectual and existential struggle, or, it should be. — Constance

For most, very little. meaning one either retreats beneath sand of old stories and rituals or one just rejects the sense of the confrontation, like Wittgenstein. — Constance

#!/usr/bin/env qjs

//it is possible to generate the solution space automatically
//but it is so small that it is easier to just supply it manually
var solutionSpace= [
			{"A":"truth", "B":"liar","C":"random"},
			{"A":"liar","B":"truth","C":"random"},
			{"A":"random","B":"truth","C":"liar"},
			{"A":"random","B":"liar","C":"truth"},
			{"A":"truth","B":"random","C":"liar"},
			{"A":"liar","B":"random","C":"truth" }
		   ];

//constraint.index is just for the purpose of reference
var constraints = [ 
	{"index":"a","who_says":"A","B_is":"truth"},
	{"index":"b","who_says":"B", "B_is":"random"},
	{"index":"c","who_says":"C", "B_is":"liar"}
	];

//we iterate over every potential solution in the solution space
for(var solution of solutionSpace) {
	var B=solution["B"];
    console.log("--------------------------");
    //A and C are just for printing the potential solution
    //they are not needed for the algorithm
	var A=solution["A"];
	var C=solution["C"];
    console.log("checking: "+A+" "+B+" "+C);
    //we assume that the solution is valid, until it isn't anymore.
    var abort_solution=false;
    //now we check every constraint for the current potential solution
    for(var constraint of constraints) {
        var index=constraint["index"];
        var who_says=constraint["who_says"];
        var B_is=constraint["B_is"];
        //check 1: truth is not allowed to lie about B
        if(solution[who_says]=="truth" && B!==B_is) {
            console.log("violation of constraint ("+index+
                "); truth is not allowed to lie and say that "+B+" is "+B_is);
            abort_solution=true;
            break;
        }
        //check 2: liar is not allowed to tell the truth about
        if(solution[who_says]=="liar" && B==B_is) {
            console.log("violation of constraint ("+index+
                "); liar is not allowed to tell the truth and say that "+B+" is "+B_is);
            abort_solution=true;
            break;
        }
        //we cannot check random; so, no check for that one
    }
    if(!abort_solution) console.log("found legitimate solution");
}
console.log("--------------------------");

$ ./truth-liar-random.js
--------------------------
checking: truth liar random
violation of constraint (a); truth is not allowed to lie and say that liar is truth
--------------------------
checking: liar truth random
violation of constraint (a); liar is not allowed to tell the truth and say that truth is truth
--------------------------
checking: random truth liar
violation of constraint (b); truth is not allowed to lie and say that truth is random
--------------------------
checking: random liar truth
found legitimate solution
--------------------------
checking: truth random liar
violation of constraint (a); truth is not allowed to lie and say that random is truth
--------------------------
checking: liar random truth
violation of constraint (c); truth is not allowed to lie and say that random is liar
--------------------------

How First Order Logic achieves this is beyond my pay grade. — RussellA

Btw, have you read Yanofsky's A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points that we discussed on another thread, should be important to this too — ssu

https://en.wikipedia.org/wiki/Cantor%27s_theorem

Theorem (Cantor) — Let f be a map from set A to its power set P ( A ). Then f : A → P ( A ) is not surjective. As a consequence, card ⁡ ( A ) < card ⁡ ( P ( A ) ) holds for any set A.

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

Theorem 1 (Cantor’s Theorem) If Y is a set and there exists a function α : Y → Y without a fixed point (for all y ∈ Y , α(y) != y), then for all sets T and for all functions f : T × T → Y there exists a function g : T → Y that is not representable by f i.e. such that for all t ∈ T: g(−) != f (−, t).

So your question, what is the true meaning of religion, is itself an expression of the basic religious impulse to fill the symbolic space. In this case, the space behind "religion".

And this is why science is a competitor to religion. Not because the mechanistic accounts of how things work differ. But because it offers a parallel, and empirically grounded, vision of what explaining the meaning of things looks like. The tree isn't just the tree we see. It is the vast scientific story that explains it. — hypericin

I was referring to real physical systems which are not conceptual — Janus

I was referring to real physical systems which are not conceptual, not I was not referring to mathematical systems, which are conceptual. — Janus

It makes no sense to say that the Universe, a real physical system, is incomplete, but of course our understanding of the universe is incomplete, and always will be. — Janus

So, the future is not comprehensively predictable, but it does not follow that it is incomplete or in possession of free will. — Janus

It's systems that are "incomplete": the idea makes no sense at all, but our understanding of systems. — Janus

It should be understood here that computers cannot follow an order of "do something else". — ssu

Ok. Oracle gives a final spoken prediction, but secretly writes down what it knows thwarter will do at that point. — Patterner

(1) Person A claims person B always tells the truth.
(2) Person B claims person B (himself) sometimes tells the truth.
(3) Person C claims person B always lies.

Or, at any point, oracle might say, "I'll (app equivalent of) write it down, and, after you act, you can read it. And you'll see I predicted accurately." — Patterner

And if you say that deep down coin flips are deterministic, so are programs. — fishfry

They... aren't that hard to avoid. You're literally not trying. — flannel jesus

Tarskian, You may be interested in a recent paper by Joel David Hamkins. [...] Terrific, readable paper. Hamkins rocks.
https://arxiv.org/pdf/2407.00680

At bottom, the logic of the argument is like this: if we had a computable way of finding whether existential statements are true, then we could iterate this with negation to also compute ∀∃ assertions, since ∀k∃n ϕ fails just in case there is some k for which the existential statement about it fails. In short, if in general existential statements are decidable, then the whole arithmetic hierarchy collapses.

And who came up with that sentence? — flannel jesus

The task given to the oracle doesn't make sense. The task given to the oracle is "predict the output of this Thw program, after you feed into the Thw program your prediction for the output of the Thw program."

It's recursive in a way that means the oracle can't even begin. — flannel jesus

Namely, assume toward contradiction that the symbol-printing problem were computably decidable, and fix a method of solving this problem. Using this as a subroutine, consider the algorithm q which on input p, a program, asks whether p on input p would ever print 0 as output. If so, then q will halt immediately without printing 0; but if not, then q prints 0 immediately as output. So q has the opposite behavior on input p with respect to printing 0 as output than p has on input p. Running q on input q will therefore print 0 as output if and only if it will not, a contradiction.

Namely, assume toward contradiction that the symbol-printing problem were computably decidable, and fix a method of solving this problem. Using the oracle as a subroutine, consider the thwarter program which asks to the oracle whether any program p on input p would ever print 0 as output. If the oracle answers that it will print 0, then thwarter itself will not print 0; but if the oracle says that thwarter doesn't print 0, then thwarter does print 0. Running thwarter on itself as input will therefore print 0 as output if and only if the oracle says that thwarter will not, a contradiction.

You don't seem interested in trying to make yourself clear, in trying to develop a self-consistent vocabulary for your ideas. You end your post with "Sometimes it still works flawlessly. Sometimes, it doesn't." as if there's nothing at all you could do to clarify your ideas.

Maybe there's not, maybe you can't clarify your ideas. — flannel jesus